september 29, 2013Œ Œ ·u^sü ˙Ÿ Œx 1;x 2;x 3;x 4;:::£‘~p‘fx ng⁄" ~x§ (i) 1; 1;1;...
TRANSCRIPT
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September 29, 2013
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ê�´U^Sü����êx1, x2, x3, x4, . . .£�~P�{xn}¤"
~X§
(I) 1,−1, 1,−1, 1,−1, 1,−1, 1,−1, 1,−1, 1 . . .(II) 1,−4, 9,−16, 25,−36, 49,−64, 81,−100, 121,−144, . . .(III) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
(IV) 1,−1
2,1
3,−1
4,1
5,−1
6,1
7,−1
8,1
9,− 1
10, . . .
(V) sinπ, 2 sinπ
2, 3 sin
π
3, 4 sin
π
4, 5 sin
π
5, 6 sin
π
6, 7 sin
π
7, 8 sin
π
8, . . .
(VI) tan 1, tan(tan 1), tan(tan(tan 1)), tan(tan(tan(tan 1))), . . .
(VII) 1, 1 +1
4, 1 +
1
4+
1
9, 1 +
1
4+
1
9+
1
16, 1 +
1
4+
1
9+
1
16+
1
25, . . .
þ¡�ê�(IV)§(V)§(VII)kù�����Ó:µê�¥���¬�5��Cu,�ê"~X§ê�(V)¥�ê¬�5��Cuπµ
10 sinπ
10∼ 3.090169944, 100 sin
π
100∼ 3.141075908,
1000 sinπ
1000∼ 3.141587486, 10000 sin
π
10000∼ 3.141592602, . . .
�H (H®�ÆêÆX) ê��4� September 29, 2013 2 / 34
ê�
ê�´U^Sü����êx1, x2, x3, x4, . . .£�~P�{xn}¤"~X§(I) 1,−1, 1,−1, 1,−1, 1,−1, 1,−1, 1,−1, 1 . . .(II) 1,−4, 9,−16, 25,−36, 49,−64, 81,−100, 121,−144, . . .(III) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
(IV) 1,−1
2,1
3,−1
4,1
5,−1
6,1
7,−1
8,1
9,− 1
10, . . .
(V) sinπ, 2 sinπ
2, 3 sin
π
3, 4 sin
π
4, 5 sin
π
5, 6 sin
π
6, 7 sin
π
7, 8 sin
π
8, . . .
(VI) tan 1, tan(tan 1), tan(tan(tan 1)), tan(tan(tan(tan 1))), . . .
(VII) 1, 1 +1
4, 1 +
1
4+
1
9, 1 +
1
4+
1
9+
1
16, 1 +
1
4+
1
9+
1
16+
1
25, . . .
þ¡�ê�(IV)§(V)§(VII)kù�����Ó:µê�¥���¬�5��Cu,�ê"~X§ê�(V)¥�ê¬�5��Cuπµ
10 sinπ
10∼ 3.090169944, 100 sin
π
100∼ 3.141075908,
1000 sinπ
1000∼ 3.141587486, 10000 sin
π
10000∼ 3.141592602, . . .
�H (H®�ÆêÆX) ê��4� September 29, 2013 2 / 34
ê�
ê�´U^Sü����êx1, x2, x3, x4, . . .£�~P�{xn}¤"~X§(I) 1,−1, 1,−1, 1,−1, 1,−1, 1,−1, 1,−1, 1 . . .(II) 1,−4, 9,−16, 25,−36, 49,−64, 81,−100, 121,−144, . . .(III) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
(IV) 1,−1
2,1
3,−1
4,1
5,−1
6,1
7,−1
8,1
9,− 1
10, . . .
(V) sinπ, 2 sinπ
2, 3 sin
π
3, 4 sin
π
4, 5 sin
π
5, 6 sin
π
6, 7 sin
π
7, 8 sin
π
8, . . .
(VI) tan 1, tan(tan 1), tan(tan(tan 1)), tan(tan(tan(tan 1))), . . .
(VII) 1, 1 +1
4, 1 +
1
4+
1
9, 1 +
1
4+
1
9+
1
16, 1 +
1
4+
1
9+
1
16+
1
25, . . .
þ¡�ê�(IV)§(V)§(VII)kù�����Ó:µê�¥���¬�5��Cu,�ê"
~X§ê�(V)¥�ê¬�5��Cuπµ
10 sinπ
10∼ 3.090169944, 100 sin
π
100∼ 3.141075908,
1000 sinπ
1000∼ 3.141587486, 10000 sin
π
10000∼ 3.141592602, . . .
�H (H®�ÆêÆX) ê��4� September 29, 2013 2 / 34
ê�
ê�´U^Sü����êx1, x2, x3, x4, . . .£�~P�{xn}¤"~X§(I) 1,−1, 1,−1, 1,−1, 1,−1, 1,−1, 1,−1, 1 . . .(II) 1,−4, 9,−16, 25,−36, 49,−64, 81,−100, 121,−144, . . .(III) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
(IV) 1,−1
2,1
3,−1
4,1
5,−1
6,1
7,−1
8,1
9,− 1
10, . . .
(V) sinπ, 2 sinπ
2, 3 sin
π
3, 4 sin
π
4, 5 sin
π
5, 6 sin
π
6, 7 sin
π
7, 8 sin
π
8, . . .
(VI) tan 1, tan(tan 1), tan(tan(tan 1)), tan(tan(tan(tan 1))), . . .
(VII) 1, 1 +1
4, 1 +
1
4+
1
9, 1 +
1
4+
1
9+
1
16, 1 +
1
4+
1
9+
1
16+
1
25, . . .
þ¡�ê�(IV)§(V)§(VII)kù�����Ó:µê�¥���¬�5��Cu,�ê"~X§ê�(V)¥�ê¬�5��Cuπµ
10 sinπ
10∼ 3.090169944, 100 sin
π
100∼ 3.141075908,
1000 sinπ
1000∼ 3.141587486, 10000 sin
π
10000∼ 3.141592602, . . .
�H (H®�ÆêÆX) ê��4� September 29, 2013 2 / 34
ê��4��ε− N�ó
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�x1, x2, x3, . . .���ê�§A�¢ê"�é?¿ε > 0§Ñ�3��g,êN = N(ε)£ùpN(ε)L«N´�εk'�¤§¦�é¤kn > N§Ñk
|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
eê�{xn}Âñu,�¢êA§K¡{xn}�Âñê�§��¡�uÑê�"Âñê��4��½´���"¯¢þ§£�y¤b�ê�{xn}Ó�Âñuü�ØÓ�¢êA1,A2©-ε=
12 |A1−A2|©@od4��½Â§�3N1§
¦�é¤kn > N1§k|xn − A1| < ε©q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2| 6 |A1−xn|+ |xn−A2| < ε+ε = |A1−A2|.gñ"
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ê��4��ε− N�ó
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�x1, x2, x3, . . .���ê�§A�¢ê"�é?¿ε > 0§Ñ�3��g,êN = N(ε)£ùpN(ε)L«N´�εk'�¤§¦�é¤kn > N§Ñk
|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
eê�{xn}Âñu,�¢êA§K¡{xn}�Âñê�§��¡�uÑê�"Âñê��4��½´���"¯¢þ§£�y¤b�ê�{xn}Ó�Âñuü�ØÓ�¢êA1,A2©-ε=
12 |A1−A2|©@od4��½Â§�3N1§
¦�é¤kn > N1§k|xn − A1| < ε©q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2| 6 |A1−xn|+ |xn−A2| < ε+ε = |A1−A2|.gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 3 / 34
ê��4��ε− N�ó
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�x1, x2, x3, . . .���ê�§A�¢ê"�é?¿ε > 0§Ñ�3��g,êN = N(ε)£ùpN(ε)L«N´�εk'�¤§¦�é¤kn > N§Ñk
|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
ê�4��ε− N�ó½Â�¢�Ò´§ÃØε > 0õo�§��nv�§@oÒk|xn − A| < ε§=xnv�CuA©
eê�{xn}Âñu,�¢êA§K¡{xn}�Âñê�§��¡�uÑê�"Âñê��4��½´���"¯¢þ§£�y¤b�ê�{xn}Ó�Âñuü�ØÓ�¢êA1,A2©-ε=
12 |A1−A2|©@od4��½Â§�3N1§
¦�é¤kn > N1§k|xn − A1| < ε©q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2| 6 |A1−xn|+ |xn−A2| < ε+ε = |A1−A2|.gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 3 / 34
ê��4��ε− N�ó
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�x1, x2, x3, . . .���ê�§A�¢ê"�é?¿ε > 0§Ñ�3��g,êN = N(ε)£ùpN(ε)L«N´�εk'�¤§¦�é¤kn > N§Ñk
|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
eê�{xn}Âñu,�¢êA§K¡{xn}�Âñê�§��¡�uÑê�"
Âñê��4��½´���"¯¢þ§£�y¤b�ê�{xn}Ó�Âñuü�ØÓ�¢êA1,A2©-ε=
12 |A1−A2|©@od4��½Â§�3N1§
¦�é¤kn > N1§k|xn − A1| < ε©q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2| 6 |A1−xn|+ |xn−A2| < ε+ε = |A1−A2|.gñ"
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ê��4��ε− N�ó
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|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
eê�{xn}Âñu,�¢êA§K¡{xn}�Âñê�§��¡�uÑê�"Âñê��4��½´���"
¯¢þ§£�y¤b�ê�{xn}Ó�Âñuü�ØÓ�¢êA1,A2©-ε=
12 |A1−A2|©@od4��½Â§�3N1§
¦�é¤kn > N1§k|xn − A1| < ε©q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2| 6 |A1−xn|+ |xn−A2| < ε+ε = |A1−A2|.gñ"
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|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
eê�{xn}Âñu,�¢êA§K¡{xn}�Âñê�§��¡�uÑê�"Âñê��4��½´���"¯¢þ§£�y¤b�ê�{xn}Ó�Âñuü�ØÓ�¢êA1,A2©
-ε=12 |A1−A2|©@od4��½Â§�3N1§
¦�é¤kn > N1§k|xn − A1| < ε©q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2| 6 |A1−xn|+ |xn−A2| < ε+ε = |A1−A2|.gñ"
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ê��4��ε− N�ó
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|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
eê�{xn}Âñu,�¢êA§K¡{xn}�Âñê�§��¡�uÑê�"Âñê��4��½´���"¯¢þ§£�y¤b�ê�{xn}Ó�Âñuü�ØÓ�¢êA1,A2©-ε=
12 |A1−A2|©
@od4��½Â§�3N1§¦�é¤kn > N1§k|xn − A1| < ε©q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2| 6 |A1−xn|+ |xn−A2| < ε+ε = |A1−A2|.gñ"
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ê��4��ε− N�ó
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|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
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12 |A1−A2|©@od4��½Â§�3N1§
¦�é¤kn > N1§k|xn − A1| < ε©
q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2| 6 |A1−xn|+ |xn−A2| < ε+ε = |A1−A2|.gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 3 / 34
ê��4��ε− N�ó
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|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
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12 |A1−A2|©@od4��½Â§�3N1§
¦�é¤kn > N1§k|xn − A1| < ε©q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©
@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2| 6 |A1−xn|+ |xn−A2| < ε+ε = |A1−A2|.gñ"
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ê��4��ε− N�ó
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�x1, x2, x3, . . .���ê�§A�¢ê"�é?¿ε > 0§Ñ�3��g,êN = N(ε)£ùpN(ε)L«N´�εk'�¤§¦�é¤kn > N§Ñk
|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
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12 |A1−A2|©@od4��½Â§�3N1§
¦�é¤kn > N1§k|xn − A1| < ε©q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2|
6 |A1−xn|+ |xn−A2| < ε+ε = |A1−A2|.gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 3 / 34
ê��4��ε− N�ó
ù«ê�¥���5��C,�ê�y�I�lêÆþî�/£ã"
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�x1, x2, x3, . . .���ê�§A�¢ê"�é?¿ε > 0§Ñ�3��g,êN = N(ε)£ùpN(ε)L«N´�εk'�¤§¦�é¤kn > N§Ñk
|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
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12 |A1−A2|©@od4��½Â§�3N1§
¦�é¤kn > N1§k|xn − A1| < ε©q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2| 6 |A1−xn|+ |xn−A2|
< ε+ε = |A1−A2|.gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 3 / 34
ê��4��ε− N�ó
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�x1, x2, x3, . . .���ê�§A�¢ê"�é?¿ε > 0§Ñ�3��g,êN = N(ε)£ùpN(ε)L«N´�εk'�¤§¦�é¤kn > N§Ñk
|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
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12 |A1−A2|©@od4��½Â§�3N1§
¦�é¤kn > N1§k|xn − A1| < ε©q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2| 6 |A1−xn|+ |xn−A2| < ε+ε = |A1−A2|.
gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 3 / 34
ê��4��ε− N�ó
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½Â
�x1, x2, x3, . . .���ê�§A�¢ê"�é?¿ε > 0§Ñ�3��g,êN = N(ε)£ùpN(ε)L«N´�εk'�¤§¦�é¤kn > N§Ñk
|xn − A| < ε.
@o·�¡ê�{xn}±¢êA�4�§½¡ê�{xn}ÂñuA§P�
limn→∞
xn = A.
eê�{xn}Âñu,�¢êA§K¡{xn}�Âñê�§��¡�uÑê�"Âñê��4��½´���"¯¢þ§£�y¤b�ê�{xn}Ó�Âñuü�ØÓ�¢êA1,A2©-ε=
12 |A1−A2|©@od4��½Â§�3N1§
¦�é¤kn > N1§k|xn − A1| < ε©q�3N2§¦�é¤kn > N2§k|xn − A2| < ε©@oén > max{N1,N2}§k|A1−A2| = |A1−xn+xn−A2| 6 |A1−xn|+ |xn−A2| < ε+ε = |A1−A2|.gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 3 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n− 0
∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©|^ε− N�óy²ê�4��§ k��½�@´"
é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§
�N =©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n− 0
∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N = ??©
@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n− 0
∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N = ??©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n− 0
∣∣∣∣ = 1
n
61
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N = ??©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n− 0
∣∣∣∣ = 1
n6
1
N + 1£��oº¤
=1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N = ??©@oé¤kn > N§Ñk£ù�·�I�
1N+1 < ε§=N + 1 > 1
婤∣∣∣∣(−1)n−1n− 0
∣∣∣∣ = 1
n6
1
N + 1
=1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1
=1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1
<11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
5¿[x ]´Ø�Lx����ê§Ïd[x ] + 1 > x©
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
dþ¡�~f��§|^ε− N�óy²ê�4��L§§Ò´é?¿�½�ε§�ÏéÙ�éA�N©
dùpÀJN =[1ε
]Òv"Ï�n�
NÑ´�ê§n > NÒ¿�Xn > N + 1 =[1ε
]+ 1 >
1ε©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
dþ¡�~f��§|^ε− N�óy²ê�4��L§§Ò´é?¿�½�ε§�ÏéÙ�éA�N©dùpÀJN =
[1ε
]Òv"
�n�
NÑ´�ê§n > NÒ¿�Xn > N + 1 =[1ε
]+ 1 >
1ε©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
dþ¡�~f��§|^ε− N�óy²ê�4��L§§Ò´é?¿�½�ε§�ÏéÙ�éA�N©dùpÀJN =
[1ε
]Òv"Ï�n�
NÑ´�ê§n > NÒ¿�Xn > N + 1 =[1ε
]+ 1 >
1ε©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
dþ¡�~f��§|^ε− N�óy²ê�4��L§§Ò´é?¿�½�ε§�ÏéÙ�éA�N©dùpÀJN =
[1ε
]Òv"Ï�n�
NÑ´�ê§n > NÒ¿�Xn > N + 1 =[1ε
]+ 1 >
1ε©
3|^ε− N�óy²�§�I��Ä@���ε§ é��ε�±Ø��Ä"
ù´Ï�XJ|xn − A| < εé���ε¤á�{§@oé���ε�\¤á"¤±k��±Ø�b�ε�u,�ê"
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
dþ¡�~f��§|^ε− N�óy²ê�4��L§§Ò´é?¿�½�ε§�ÏéÙ�éA�N©dùpÀJN =
[1ε
]Òv"Ï�n�
NÑ´�ê§n > NÒ¿�Xn > N + 1 =[1ε
]+ 1 >
1ε©
3|^ε− N�óy²�§�I��Ä@���ε§ é��ε�±Ø��Ä"ù´Ï�XJ|xn − A| < εé���ε¤á�{§@oé���ε�\¤á"
¤±k��±Ø�b�ε�u,�ê"
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
dþ¡�~f��§|^ε− N�óy²ê�4��L§§Ò´é?¿�½�ε§�ÏéÙ�éA�N©dùpÀJN =
[1ε
]Òv"Ï�n�
NÑ´�ê§n > NÒ¿�Xn > N + 1 =[1ε
]+ 1 >
1ε©
3|^ε− N�óy²�§�I��Ä@���ε§ é��ε�±Ø��Ä"ù´Ï�XJ|xn − A| < εé���ε¤á�{§@oé���ε�\¤á"¤±k��±Ø�b�ε�u,�ê"
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
3|^ε− N�óy²�§�I��Ä@���ε§ é��ε�±Ø��Ä"ù´Ï�XJ|xn − A| < εé���ε¤á�{§@oé���ε�\¤á"¤±k��±Ø�b�ε�u,�ê"
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©
�N =©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =??©
Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =??©Ké¤kn>N§Ñk
|qn − 0| = |q|n
6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N =??©Ké¤kn>N§Ñk£ù�I�|q|n < ε§=n > log|q| 婤
|qn − 0| = |q|n
6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N = [log|q| ε]©Ké¤kn>N§Ñk
|qn − 0| = |q|n
6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N = [log|q| ε]©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1
= |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N = [log|q| ε]©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1
< |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y²ê�{(−1)n−1 1n}Âñu0§= limn→∞
(−1)n−1
n= 0.
y©é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣(−1)n−1n
− 0∣∣∣∣ = 1
n6
1
N + 1=
1
[1ε ] + 1<
11ε
= ε.
Ïdd4��½Â§{(−1)n−1 1n}Âñu0©
~
�|q| < 1©y² limn→∞
qn = 0.
y©é?¿ε> 0§Ø�b�ε< 1©�N = [log|q| ε]©Ké¤kn>N§Ñk
|qn − 0| = |q|n 6 |q|N+1 = |q|[log|q| ε]+1 < |q|log|q| ε = ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 4 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N =©@oé¤kn>N§Ñk∣∣∣∣100nn!− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
3|^ε− N�óy²�§¢Sþ���´N��35" é�½�姤��N´Ä(¢´���ÀJ§K¿Ø�½´7L�"
¤±k� Ø�rN�����"
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N =©@oé¤kn>N§Ñk∣∣∣∣100nn!− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
3|^ε− N�óy²�§¢Sþ���´N��35" é�½�姤��N´Ä(¢´���ÀJ§K¿Ø�½´7L�"¤±k� Ø�rN�����"
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N =©@oé¤kn>N§Ñk∣∣∣∣100nn!− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
3|^ε− N�óy²�§¢Sþ���´N��35" é�½�姤��N´Ä(¢´���ÀJ§K¿Ø�½´7L�"¤±k� Ø�rN�����"
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n
6100100
100!· 100
n.
@oé?¿ε> 0§�N =©@oé¤kn>N§Ñk∣∣∣∣100nn!− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
3|^ε− N�óy²�§¢Sþ���´N��35" é�½�姤��N´Ä(¢´���ÀJ§K¿Ø�½´7L�"¤±k� Ø�rN�����"
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n
6100100
100!· 100
n.
@oé?¿ε> 0§�N =©@oé¤kn>N§Ñk∣∣∣∣100nn!− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
3|^ε− N�óy²�§¢Sþ���´N��35" é�½�姤��N´Ä(¢´���ÀJ§K¿Ø�½´7L�"¤±k� Ø�rN�����"
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N =??©@oé¤kn>N§Ñk∣∣∣∣100nn!− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
3|^ε− N�óy²�§¢Sþ���´N��35" é�½�姤��N´Ä(¢´���ÀJ§K¿Ø�½´7L�"¤±k� Ø�rN�����"
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N =??©
@oé¤kn>N§Ñk∣∣∣∣100nn!− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
3|^ε− N�óy²�§¢Sþ���´N��35" é�½�姤��N´Ä(¢´���ÀJ§K¿Ø�½´7L�"¤±k� Ø�rN�����"
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N =??©@oé¤kn>N§Ñk∣∣∣∣100nn!− 0
∣∣∣∣ = 100n
n!
6100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
3|^ε− N�óy²�§¢Sþ���´N��35" é�½�姤��N´Ä(¢´���ÀJ§K¿Ø�½´7L�"¤±k� Ø�rN�����"
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N =??©@oé¤kn>N§Ñk∣∣∣∣100nn!− 0
∣∣∣∣ = 100n
n!6
100101
100!n
6100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
3|^ε− N�óy²�§¢Sþ���´N��35" é�½�姤��N´Ä(¢´���ÀJ§K¿Ø�½´7L�"¤±k� Ø�rN�����"
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N =??©@oé¤kn>N§Ñk
£ù���100101
100!n < ε§�Ò´n > 100101
100!ε =�"¤∣∣∣∣100nn!− 0
∣∣∣∣ = 100n
n!6
100101
100!n
6100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
3|^ε− N�óy²�§¢Sþ���´N��35" é�½�姤��N´Ä(¢´���ÀJ§K¿Ø�½´7L�"¤±k� Ø�rN�����"
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N = max{100,[100101100!ε
]}©@oé¤kn>N§Ñk∣∣∣∣100nn!
− 0
∣∣∣∣ = 100n
n!6
100101
100!n
6100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
3|^ε− N�óy²�§¢Sþ���´N��35" é�½�姤��N´Ä(¢´���ÀJ§K¿Ø�½´7L�"¤±k� Ø�rN�����"
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N = max{100,[100101100!ε
]}©@oé¤kn>N§Ñk∣∣∣∣100nn!
− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)
<100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
3|^ε− N�óy²�§¢Sþ���´N��35" é�½�姤��N´Ä(¢´���ÀJ§K¿Ø�½´7L�"¤±k� Ø�rN�����"
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N = max{100,[100101100!ε
]}©@oé¤kn>N§Ñk∣∣∣∣100nn!
− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N = ??©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N = max{100,[100101100!ε
]}©@oé¤kn>N§Ñk∣∣∣∣100nn!
− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N = ??©
@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N = max{100,[100101100!ε
]}©@oé¤kn>N§Ñk∣∣∣∣100nn!
− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N = ??©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣
=2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N = max{100,[100101100!ε
]}©@oé¤kn>N§Ñk∣∣∣∣100nn!
− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N = ??©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N = max{100,[100101100!ε
]}©@oé¤kn>N§Ñk∣∣∣∣100nn!
− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N = ??©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)5¿�2n − 5 < 2n − 4
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N = max{100,[100101100!ε
]}©@oé¤kn>N§Ñk∣∣∣∣100nn!
− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N = ??©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)6
n − 2
2n2 − 4n=
1
2n
5¿�2n2 − 3 > 2n2 − 4n
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N = max{100,[100101100!ε
]}©@oé¤kn>N§Ñk∣∣∣∣100nn!
− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N = ??©@oé¤kn > N§Ñk
£ù��� 12n < ε§�Ò´n > 1
2ε=�"¤∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)6
n − 2
2n2 − 4n=
1
2n�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N = max{100,[100101100!ε
]}©@oé¤kn>N§Ñk∣∣∣∣100nn!
− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N = max{2,[ 12ε
]}©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)6
n − 2
2n2 − 4n=
1
2n
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N = max{100,[100101100!ε
]}©@oé¤kn>N§Ñk∣∣∣∣100nn!
− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N = max{2,[ 12ε
]}©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)6
n − 2
2n2 − 4n=
1
2n6
1
2(N + 1)
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
100n
n!= 0 (2) lim
n→∞n2 + n − 4
2n2 − 3=
1
2.
y©(1) w,XJn > 100§k
100n
n!=
100100
100!· 100101· 100102· · · · · 100
n6
100100
100!· 100
n.
@oé?¿ε> 0§�N = max{100,[100101100!ε
]}©@oé¤kn>N§Ñk∣∣∣∣100nn!
− 0
∣∣∣∣ = 100n
n!6
100101
100!n6
100101
100!(N + 1)<
100101
100! · 100101100!ε
= ε.
(2) é?¿ε > 0§�N = max{2,[ 12ε
]}©@oé¤kn > N§Ñk∣∣∣∣n2 + n − 4
2n2 − 3− 1
2
∣∣∣∣ =∣∣∣∣2(n2 + n − 4)− (2n2 − 3)
2(2n2 − 3)
∣∣∣∣ = 2n − 5
2(2n2 − 3)
62(n − 2)
2(2n2 − 3)6
n − 2
2n2 − 4n=
1
2n6
1
2(N + 1)<
1
2 · 12ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 5 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n
=1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =©@oé¤kn > N§Ñk∣∣∣∣√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N = ??©
@oé¤kn > N§Ñk∣∣∣∣√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N = ??©@oé¤kn > N§Ñk∣∣∣∣√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n
61
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N = ??©@oé¤kn > N§Ñk£ù��� 1
n < ε=�"¤∣∣∣∣√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n
61
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n
61
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1
<11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n
=n∑
k=0
(n
k
)é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)|Üê
(n
k
)=
n!
k!(n − k)!§�k�Ø�5��P{§XC k
n§C (n, k)�"
é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)>(n
2
)
é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)>(n
2
)=
n!
2! · (n − 2)!
é?¿ε > 0§�N =©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)>(n
2
)=
n!
2! · (n − 2)!=
1
2n(n − 1).
é?¿ε > 0§�N = ??©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)>(n
2
)=
n!
2! · (n − 2)!=
1
2n(n − 1).
é?¿ε > 0§�N = ??©
Ké¤kn > N§Ñk∣∣∣∣ n2n − 0∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)>(n
2
)=
n!
2! · (n − 2)!=
1
2n(n − 1).
é?¿ε > 0§�N = ??©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0∣∣∣∣ = n
2n
6n
12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)>(n
2
)=
n!
2! · (n − 2)!=
1
2n(n − 1).
é?¿ε > 0§�N = ??©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)>(n
2
)=
n!
2! · (n − 2)!=
1
2n(n − 1).
é?¿ε > 0§�N = ??©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 1
62
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)>(n
2
)=
n!
2! · (n − 2)!=
1
2n(n − 1).
é?¿ε > 0§�N = ??©Ké¤kn > N§Ñk £�� 2n−1 < ε=�"¤∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 1
62
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)>(n
2
)=
n!
2! · (n − 2)!=
1
2n(n − 1).
é?¿ε > 0§�N =[2ε
]+ 1©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 1
62
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)>(n
2
)=
n!
2! · (n − 2)!=
1
2n(n − 1).
é?¿ε > 0§�N =[2ε
]+ 1©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N
<22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
y² (1) limn→∞
√n2 − 1
n= 1. (2) lim
n→∞n
2n= 0.
y©(1) ·�k
1−√n2 − 1
n=
n −√n2 − 1
n=
1
n(n +√n2 − 1)
61
n.
é?¿ε > 0§�N =[1ε
]©@oé¤kn > N§Ñk∣∣∣∣
√n2 − 1
n− 1
∣∣∣∣ = 1−√n2 − 1
n6
1
n6
1
N + 1<
11ε
= ε.
(2) |^��ª½n§éun > 2§·�k
2n = (1 + 1)n =n∑
k=0
(n
k
)>(n
2
)=
n!
2! · (n − 2)!=
1
2n(n − 1).
é?¿ε > 0§�N =[2ε
]+ 1©Ké¤kn > N§Ñk∣∣∣∣ n2n − 0
∣∣∣∣ = n
2n6
n12n(n − 1)
=2
n − 16
2
N<
22ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 6 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©
@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n
=n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n
=n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk
>(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3
=α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =??©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =??©
Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =??©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n
6n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =??©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =??©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)5¿
n
n − 1≤ 3
2
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =??©Ké¤kn > N§k£��9
α3(n−2) <ε=�"¤
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =[ 9α3ε
]+2©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =[ 9α3ε
]+2©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)6
9
α3(N − 1)
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
�|q| < 1§y² limn→∞
n2qn = 0©
y©-α =1|q| − 1©@o
|n2qn| = n2
( 1|q|)
n=
n2
(1 + α)n.
d��ª½n§éun > 3§·�k
(1 + α)n =n∑
k=0
(n
k
)αk >
(n
3
)α3 =
n!
3! · (n − 3)!α3 =
α3
6n(n − 1)(n − 2).
é?¿ε> 0§�N =[ 9α3ε
]+2©Ké¤kn > N§k
|n2qn − 0| = n2
(1 + α)n6
n2
α3
6 n(n − 1)(n − 2)=
6n
α3(n − 1)(n − 2)
69
α3(n − 2)6
9
α3(N − 1)<
9
α3 · 12α3ε
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 7 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©
£�y¤� n√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ"
é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©
@oén > N§k 1n 6 1
N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) ?nmng��¯K�§~�±¦^�ªxn − 1 = (x − 1)(xn−1 + xn−2 + · · ·+ x + 1),
=
x − 1 =xn − 1
xn−1 + xn−2 + · · ·+ x + 1.
(2) én > 3§
-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©
Km > n−12 > n
3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©
5¿n − 1
2>
n
3⇐⇒ n > 3.
@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1
=(n
12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
5¿ùpn2m−12m + · · ·+ n
m+12m + n
m2m 6 n
2m−12m + · · ·+ n
12m + 1
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
6n
m · nm2m
=n
m ·√n
n2m−12m , . . . , n
m+12m , n
m2m��m�ê§Ù¥���´n
m2m©
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
6n
m · nm2m
=n
m ·√n6
3√n.
é?¿ε > 0§�N =©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
6n
m · nm2m
=n
m ·√n6
3√n.
é?¿ε > 0§�N = ??©
Kén > N§k| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
6n
m · nm2m
=n
m ·√n6
3√n.
é?¿ε > 0§�N = ??©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n
63√
N + 1<
3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
6n
m · nm2m
=n
m ·√n6
3√n.
é?¿ε > 0§�N = ??©Kén > N§k| n√n − 1| = n
√n − 1 6
3√n£5¿
3√n< ε⇐⇒ n >
9
ε2©¤
63√
N + 1<
3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
6n
m · nm2m
=n
m ·√n6
3√n.
é?¿ε > 0§�N = max{2,[ 9ε2]}©Kén > N§k
| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
ê�4��~f
~
y² (1) limn→∞
n√3 = 1. (2) lim
n→∞n√n = 1.
y©(1) é?¿a> 1§Ñk n√a> 1©£�y¤� n
√a< 1§Ka=( n
√a)n< 1§
gñ" é?¿ε > 0§Ï�log3 x´î�4O¼ê§¤±n√3− 1 < ε⇐⇒ 3
1n < 1 + ε⇐⇒ 1
n < log3(1 + ε).
Ïd�±�N =[ 1log3(1+ε)
]©@oén > N§k 1
n 6 1N+1 < log3(1 + ε)©
=| n√3− 1| = n
√3− 1 < ε©
(2) én > 3§-m =[n2
]©Km > n−1
2 > n3©@o
n1n − 1 6 n
12m − 1 =
(n12m )2m − 1
(n12m )2m−1 + (n
12m )2m−2 + · · ·+ n
12m + 1
6n − 1
n2m−12m + · · ·+ n
m+12m + n
m2m
6n
m · nm2m
=n
m ·√n6
3√n.
é?¿ε > 0§�N = max{2,[ 9ε2]}©Kén > N§k
| n√n − 1| = n
√n − 1 6
3√n6
3√N + 1
<3√9ε−2
= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 8 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©
d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1©
-M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©
w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
5¿§þ¡½n¥�(ii)§XJ{xn}ÂñuA§=¦é¤knÑkxn > 0§
�ØU�ÑA > 0©¯¢þ§��{ü�~fÒ´ limn→∞
1
n= 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© £|xn − A| < A2�duA− A
2 < x < A+ A2¤
(ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©
K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §
=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©
ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
�H (H®�ÆêÆX) ê��4� September 29, 2013 9 / 34
Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©
dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
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Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©
=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
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Âñ5§k.5��Ò5
½n
Âñê�{xn}7k.§=�3M > 0§¦�é¤kn§Ñk|xn| 6 M©
y©� limn→∞
xn =A©d4��½Â§�3N§é¤kn>NÑk|xn − A|< 1§
=|xn| 6 |A|+ |xn − A| < |A|+ 1© -M =max{|x1|, |x2|, . . . , |xN |, |A|+ 1}©w,é¤kn§Ñk|xn| 6 M©
½n
�ê�{xn}ÂñuA©(i) eA > 0§K�3N0§é¤kn>N0§kxn > 0©(ii) e�3N0§é¤kn > N0§xn > 0§KA > 0©
y©(i) Ï�A > 0§d4�½Â§ �3N§é¤kn > N§|xn − A| < A2§
=xn>A− A2 = A
2 > 0© (ii) £�y¤�A< 0©K�3N§é¤kn>N§
|xn−A|< |A|2 §=xn<A+|A|2 = A
2 < 0©ù�én>N0kxn > 0gñ"
þ¡�½n¥�>�>�±�¤<�6©dùp�0���¤?¿¢êα©=A > α =⇒ é¿©��nkxn > α¶é¿©��nkxn > α =⇒ A > α©
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Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣ = ∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣6
|x1 − A|+ · · ·+ |xN0 − A|n
+|xN0+1 − A|+ · · ·+ |xn − A|
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2<
ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©
é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣ = ∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣6
|x1 − A|+ · · ·+ |xN0 − A|n
+|xN0+1 − A|+ · · ·+ |xn − A|
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2<
ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣ = ∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣6
|x1 − A|+ · · ·+ |xN0 − A|n
+|xN0+1 − A|+ · · ·+ |xn − A|
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2<
ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©
@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣ = ∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣6
|x1 − A|+ · · ·+ |xN0 − A|n
+|xN0+1 − A|+ · · ·+ |xn − A|
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2<
ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣
=∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣6
|x1 − A|+ · · ·+ |xN0 − A|n
+|xN0+1 − A|+ · · ·+ |xn − A|
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2<
ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣ = ∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣
6|x1 − A|+ · · ·+ |xN0 − A|
n+|xN0+1 − A|+ · · ·+ |xn − A|
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2<
ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣ = ∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣6
|x1 − A|+ · · ·+ |xN0 − A|n
+|xN0+1 − A|+ · · ·+ |xn − A|
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2<
ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣ = ∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣6
|x1 − A|+ · · ·+ |xN0 − A|n
+|xN0+1 − A|+ · · ·+ |xn − A|
n
6(|x1|+ |A|) + · · ·+ (|xN0 |+ |A|)
n+
ε2 + · · ·+ ε
2
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2<
ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣ = ∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣6
|x1 − A|+ · · ·+ |xN0 − A|n
+|xN0+1 − A|+ · · ·+ |xn − A|
n
6(|x1|+ |A|) + · · ·+ (|xN0 |+ |A|)
n+
ε2 + · · ·+ ε
2
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2
<ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣ = ∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣6
|x1 − A|+ · · ·+ |xN0 − A|n
+|xN0+1 − A|+ · · ·+ |xn − A|
n
6(|x1|+ |A|) + · · ·+ (|xN0 |+ |A|)
n+
ε2 + · · ·+ ε
2
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2<
ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣ = ∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣6
|x1 − A|+ · · ·+ |xN0 − A|n
+|xN0+1 − A|+ · · ·+ |xn − A|
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2<
ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣ = ∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣6
|x1 − A|+ · · ·+ |xN0 − A|n
+|xN0+1 − A|+ · · ·+ |xn − A|
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2<
ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
Âñê�k.5�A^
~
�ê�{xn}Âñ§Ù4��A©-yn = 1n (x1 + x2 + · · ·+ xn)©y²{yn}
�4���A©
y©Ï�{xn}Âñ§¤±k.§=�3M > 0§é¤kn§|xn| < M©é?
¿ε> 0§Ï� limn→∞
xn =A§��3N0§¦�é¤kn>N0k|xn − A|< 12 ε©
�N = max{N0,
[2(M+|A|)N0
ε
]}©@oé¤kn > N§k
|yn − A| =∣∣∣∣x1 + · · ·+ xn
n− A
∣∣∣∣ = ∣∣∣∣(x1 − A) + · · ·+ (xn − A)
n
∣∣∣∣6
|x1 − A|+ · · ·+ |xN0 − A|n
+|xN0+1 − A|+ · · ·+ |xn − A|
n
6N0 · (M + |A|)
N + 1+
n − N0
n· ε2<
ε
2+
ε
2= ε.
|^±þ(J±9ln x�ëY5£= limn→∞
ln xn = ln(limn→∞
xn)¤§�±y²µ
e¤kxn> 0§{xn}ÂñuA> 0§Kéyn = n√x1 · · · xn§k{yn}�ÂñuA©
ù´Ï�ln yn = 1n (ln x1 + · · ·+ ln xn)§Ïd{ln yn}�{ln xn}ÑÂñulnA©
�H (H®�ÆêÆX) ê��4� September 29, 2013 10 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε© -N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|6|xn−A|+ |yn−B|< ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB| 6 |(xn − A)yn|+ |A(yn − B)|<
ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©
é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε© -N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|6|xn−A|+ |yn−B|< ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB| 6 |(xn − A)yn|+ |A(yn − B)|<
ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε©
-N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|6|xn−A|+ |yn−B|< ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB| 6 |(xn − A)yn|+ |A(yn − B)|<
ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε© -N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|6|xn−A|+ |yn−B|< ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB| 6 |(xn − A)yn|+ |A(yn − B)|<
ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε© -N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|
6|xn−A|+ |yn−B|< ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB| 6 |(xn − A)yn|+ |A(yn − B)|<
ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε© -N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|6|xn−A|+ |yn−B|
<ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB| 6 |(xn − A)yn|+ |A(yn − B)|<
ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε© -N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|6|xn−A|+ |yn−B|< ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB| 6 |(xn − A)yn|+ |A(yn − B)|<
ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε© -N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|6|xn−A|+ |yn−B|< ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©
{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB| 6 |(xn − A)yn|+ |A(yn − B)|<
ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε© -N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|6|xn−A|+ |yn−B|< ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©
�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB| 6 |(xn − A)yn|+ |A(yn − B)|<
ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε© -N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|6|xn−A|+ |yn−B|< ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©
K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB| 6 |(xn − A)yn|+ |A(yn − B)|<
ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε© -N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|6|xn−A|+ |yn−B|< ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB|
6 |(xn − A)yn|+ |A(yn − B)|<
ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε© -N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|6|xn−A|+ |yn−B|< ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB| 6 |(xn − A)yn|+ |A(yn − B)|
<ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
½n
�{xn}§{yn}�ê�§ limn→∞
xn = A§ limn→∞
yn = B§A,B ∈ R©K
limn→∞
(xn ± yn) = A± B, limn→∞
(xn · yn) = A · B, limn→∞
xnyn
=A
B(eB 6= 0).
y©ky limn→∞
(xn + yn) = A+ B©é?¿ε > 0§�3N1,N2§én > N1k
|xn − A|< 12 ε¶én>N2§|yn − B|< 1
2 ε© -N =max{N1,N2}§én>N§
|(xn + yn)− (A+ B)|=|(xn−A)+ (yn−B)|6|xn−A|+ |yn−B|< ε
2+ε
2= ε.
2y limn→∞
(xn · yn)=A · B©{yn}Âñ§�k.§=�3M > 0§é¤kn§
|yn| < M©�ε > 0©�3N1,N2§én > N1§|xn − A| < ε
2M¶én > N2§
k|yn − B| < ε
2(|A|+ 1)©K�N = max{N1,N2}§é¤kn > N§
|xnyn − AB| = |xnyn − Ayn + Ayn − AB| 6 |(xn − A)yn|+ |A(yn − B)|<
ε
2M·M + |A| · ε
2(|A|+ 1)6
ε
2+ε
2= ε.
�H (H®�ÆêÆX) ê��4� September 29, 2013 11 / 34
4��oK$�
��y² limn→∞
xnyn
=A
B©
Ø��B > 0©K�3N1§én > N1Ñkyn >B
2©
é?¿ε > 0§�3N2,N3§én > N2§k|xn − A| < εB
4¶ én > N3§k
|yn − B| < εB2
4(|A|+ 1)©Ké¤kn > max{N1,N2,N3}§k∣∣∣∣xnyn − A
B
∣∣∣∣ = ∣∣∣∣xnB − AynynB
∣∣∣∣ 6 |(xn − A)B|+ |A(B − yn)||ynB|
<
εB4 · B + |A| · εB2
4(|A|+1)
|B2 · B|6
ε
2+ε
2= ε.
·�5w�e4�oK$��~f§
limn→∞
2n3 + 4n√n + 6n + 8
3n3 + 5n2 + 7√n + 9
= limn→∞
2 + 4n√n+ 6
n2+ 8
n3
3 + 5n + 7
n2√n+ 9
n3
=2 + lim
n→∞4
n√n+ lim
n→∞6n2
+ limn→∞
8n3
3 + limn→∞
5n + lim
n→∞7
n2√n+ lim
n→∞9n3
=2 + 0 + 0 + 0
3 + 0 + 0 + 0=
2
3.
�H (H®�ÆêÆX) ê��4� September 29, 2013 12 / 34
4��oK$�
��y² limn→∞
xnyn
=A
B©Ø��B > 0©
K�3N1§én > N1Ñkyn >B
2©
é?¿ε > 0§�3N2,N3§én > N2§k|xn − A| < εB
4¶ én > N3§k
|yn − B| < εB2
4(|A|+ 1)©Ké¤kn > max{N1,N2,N3}§k∣∣∣∣xnyn − A
B
∣∣∣∣ = ∣∣∣∣xnB − AynynB
∣∣∣∣ 6 |(xn − A)B|+ |A(B − yn)||ynB|
<
εB4 · B + |A| · εB2
4(|A|+1)
|B2 · B|6
ε
2+ε
2= ε.
·�5w�e4�oK$��~f§
limn→∞
2n3 + 4n√n + 6n + 8
3n3 + 5n2 + 7√n + 9
= limn→∞
2 + 4n√n+ 6
n2+ 8
n3
3 + 5n + 7
n2√n+ 9
n3
=2 + lim
n→∞4
n√n+ lim
n→∞6n2
+ limn→∞
8n3
3 + limn→∞
5n + lim
n→∞7
n2√n+ lim
n→∞9n3
=2 + 0 + 0 + 0
3 + 0 + 0 + 0=
2
3.
�H (H®�ÆêÆX) ê��4� September 29, 2013 12 / 34
4��oK$�
��y² limn→∞
xnyn
=A
B©Ø��B > 0©K�3N1§én > N1Ñkyn >
B
2©
|yn − B| < B
2=⇒ yn >
B
2
é?¿ε > 0§�3N2,N3§én > N2§k|xn − A| < εB
4¶ én > N3§k
|yn − B| < εB2
4(|A|+ 1)©Ké¤kn > max{N1,N2,N3}§k∣∣∣∣xnyn − A
B
∣∣∣∣ = ∣∣∣∣xnB − AynynB
∣∣∣∣ 6 |(xn − A)B|+ |A(B − yn)||ynB|
<
εB4 · B + |A| · εB2
4(|A|+1)
|B2 · B|6
ε
2+ε
2= ε.
·�5w�e4�oK$��~f§
limn→∞
2n3 + 4n√n + 6n + 8
3n3 + 5n2 + 7√n + 9
= limn→∞
2 + 4n√n+ 6
n2+ 8
n3
3 + 5n + 7
n2√n+ 9
n3
=2 + lim
n→∞4
n√n+ lim
n→∞6n2
+ limn→∞
8n3
3 + limn→∞
5n + lim
n→∞7
n2√n+ lim
n→∞9n3
=2 + 0 + 0 + 0
3 + 0 + 0 + 0=
2
3.
�H (H®�ÆêÆX) ê��4� September 29, 2013 12 / 34
4��oK$�
��y² limn→∞
xnyn
=A
B©Ø��B > 0©K�3N1§én > N1Ñkyn >
B
2©
é?¿ε > 0§�3N2,N3§én > N2§k|xn − A| < εB
4¶ én > N3§k
|yn − B| < εB2
4(|A|+ 1)©
Ké¤kn > max{N1,N2,N3}§k∣∣∣∣xnyn − A
B
∣∣∣∣ = ∣∣∣∣xnB − AynynB
∣∣∣∣ 6 |(xn − A)B|+ |A(B − yn)||ynB|
<
εB4 · B + |A| · εB2
4(|A|+1)
|B2 · B|6
ε
2+ε
2= ε.
·�5w�e4�oK$��~f§
limn→∞
2n3 + 4n√n + 6n + 8
3n3 + 5n2 + 7√n + 9
= limn→∞
2 + 4n√n+ 6
n2+ 8
n3
3 + 5n + 7
n2√n+ 9
n3
=2 + lim
n→∞4
n√n+ lim
n→∞6n2
+ limn→∞
8n3
3 + limn→∞
5n + lim
n→∞7
n2√n+ lim
n→∞9n3
=2 + 0 + 0 + 0
3 + 0 + 0 + 0=
2
3.
�H (H®�ÆêÆX) ê��4� September 29, 2013 12 / 34
4��oK$�
��y² limn→∞
xnyn
=A
B©Ø��B > 0©K�3N1§én > N1Ñkyn >
B
2©
é?¿ε > 0§�3N2,N3§én > N2§k|xn − A| < εB
4¶ én > N3§k
|yn − B| < εB2
4(|A|+ 1)©Ké¤kn > max{N1,N2,N3}§k∣∣∣∣xnyn − A
B
∣∣∣∣ = ∣∣∣∣xnB − AynynB
∣∣∣∣
6|(xn − A)B|+ |A(B − yn)|
|ynB|
<
εB4 · B + |A| · εB2
4(|A|+1)
|B2 · B|6
ε
2+ε
2= ε.
·�5w�e4�oK$��~f§
limn→∞
2n3 + 4n√n + 6n + 8
3n3 + 5n2 + 7√n + 9
= limn→∞
2 + 4n√n+ 6
n2+ 8
n3
3 + 5n + 7
n2√n+ 9
n3
=2 + lim
n→∞4
n√n+ lim
n→∞6n2
+ limn→∞
8n3
3 + limn→∞
5n + lim
n→∞7
n2√n+ lim
n→∞9n3
=2 + 0 + 0 + 0
3 + 0 + 0 + 0=
2
3.
�H (H®�ÆêÆX) ê��4� September 29, 2013 12 / 34
4��oK$�
��y² limn→∞
xnyn
=A
B©Ø��B > 0©K�3N1§én > N1Ñkyn >
B
2©
é?¿ε > 0§�3N2,N3§én > N2§k|xn − A| < εB
4¶ én > N3§k
|yn − B| < εB2
4(|A|+ 1)©Ké¤kn > max{N1,N2,N3}§k∣∣∣∣xnyn − A
B
∣∣∣∣ = ∣∣∣∣xnB − AynynB
∣∣∣∣ 6 |(xn − A)B|+ |A(B − yn)||ynB|
<
εB4 · B + |A| · εB2
4(|A|+1)
|B2 · B|6
ε
2+ε
2= ε.
·�5w�e4�oK$��~f§
limn→∞
2n3 + 4n√n + 6n + 8
3n3 + 5n2 + 7√n + 9
= limn→∞
2 + 4n√n+ 6
n2+ 8
n3
3 + 5n + 7
n2√n+ 9
n3
=2 + lim
n→∞4
n√n+ lim
n→∞6n2
+ limn→∞
8n3
3 + limn→∞
5n + lim
n→∞7
n2√n+ lim
n→∞9n3
=2 + 0 + 0 + 0
3 + 0 + 0 + 0=
2
3.
�H (H®�ÆêÆX) ê��4� September 29, 2013 12 / 34
4��oK$�
��y² limn→∞
xnyn
=A
B©Ø��B > 0©K�3N1§én > N1Ñkyn >
B
2©
é?¿ε > 0§�3N2,N3§én > N2§k|xn − A| < εB
4¶ én > N3§k
|yn − B| < εB2
4(|A|+ 1)©Ké¤kn > max{N1,N2,N3}§k∣∣∣∣xnyn − A
B
∣∣∣∣ = ∣∣∣∣xnB − AynynB
∣∣∣∣ 6 |(xn − A)B|+ |A(B − yn)||ynB|
<
εB4 · B + |A| · εB2
4(|A|+1)
|B2 · B|6
ε
2+ε
2= ε.
·�5w�e4�oK$��~f§
limn→∞
2n3 + 4n√n + 6n + 8
3n3 + 5n2 + 7√n + 9
= limn→∞
2 + 4n√n+ 6
n2+ 8
n3
3 + 5n + 7
n2√n+ 9
n3
=2 + lim
n→∞4
n√n+ lim
n→∞6n2
+ limn→∞
8n3
3 + limn→∞
5n + lim
n→∞7
n2√n+ lim
n→∞9n3
=2 + 0 + 0 + 0
3 + 0 + 0 + 0=
2
3.
�H (H®�ÆêÆX) ê��4� September 29, 2013 12 / 34
4��oK$�
��y² limn→∞
xnyn
=A
B©Ø��B > 0©K�3N1§én > N1Ñkyn >
B
2©
é?¿ε > 0§�3N2,N3§én > N2§k|xn − A| < εB
4¶ én > N3§k
|yn − B| < εB2
4(|A|+ 1)©Ké¤kn > max{N1,N2,N3}§k∣∣∣∣xnyn − A
B
∣∣∣∣ = ∣∣∣∣xnB − AynynB
∣∣∣∣ 6 |(xn − A)B|+ |A(B − yn)||ynB|
<
εB4 · B + |A| · εB2
4(|A|+1)
|B2 · B|6
ε
2+ε
2= ε.
·�5w�e4�oK$��~f§
limn→∞
2n3 + 4n√n + 6n + 8
3n3 + 5n2 + 7√n + 9
= limn→∞
2 + 4n√n+ 6
n2+ 8
n3
3 + 5n + 7
n2√n+ 9
n3
=2 + lim
n→∞4
n√n+ lim
n→∞6n2
+ limn→∞
8n3
3 + limn→∞
5n + lim
n→∞7
n2√n+ lim
n→∞9n3
=2 + 0 + 0 + 0
3 + 0 + 0 + 0=
2
3.
�H (H®�ÆêÆX) ê��4� September 29, 2013 12 / 34
4��oK$�
��y² limn→∞
xnyn
=A
B©Ø��B > 0©K�3N1§én > N1Ñkyn >
B
2©
é?¿ε > 0§�3N2,N3§én > N2§k|xn − A| < εB
4¶ én > N3§k
|yn − B| < εB2
4(|A|+ 1)©Ké¤kn > max{N1,N2,N3}§k∣∣∣∣xnyn − A
B
∣∣∣∣ = ∣∣∣∣xnB − AynynB
∣∣∣∣ 6 |(xn − A)B|+ |A(B − yn)||ynB|
<
εB4 · B + |A| · εB2
4(|A|+1)
|B2 · B|6
ε
2+ε
2= ε.
·�5w�e4�oK$��~f§
limn→∞
2n3 + 4n√n + 6n + 8
3n3 + 5n2 + 7√n + 9
= limn→∞
2 + 4n√n+ 6
n2+ 8
n3
3 + 5n + 7
n2√n+ 9
n3
=2 + lim
n→∞4
n√n+ lim
n→∞6n2
+ limn→∞
8n3
3 + limn→∞
5n + lim
n→∞7
n2√n+ lim
n→∞9n3
=2 + 0 + 0 + 0
3 + 0 + 0 + 0=
2
3.
�H (H®�ÆêÆX) ê��4� September 29, 2013 12 / 34
4��oK$�
��y² limn→∞
xnyn
=A
B©Ø��B > 0©K�3N1§én > N1Ñkyn >
B
2©
é?¿ε > 0§�3N2,N3§én > N2§k|xn − A| < εB
4¶ én > N3§k
|yn − B| < εB2
4(|A|+ 1)©Ké¤kn > max{N1,N2,N3}§k∣∣∣∣xnyn − A
B
∣∣∣∣ = ∣∣∣∣xnB − AynynB
∣∣∣∣ 6 |(xn − A)B|+ |A(B − yn)||ynB|
<
εB4 · B + |A| · εB2
4(|A|+1)
|B2 · B|6
ε
2+ε
2= ε.
·�5w�e4�oK$��~f§
limn→∞
2n3 + 4n√n + 6n + 8
3n3 + 5n2 + 7√n + 9
= limn→∞
2 + 4n√n+ 6
n2+ 8
n3
3 + 5n + 7
n2√n+ 9
n3
=2 + lim
n→∞4
n√n+ lim
n→∞6n2
+ limn→∞
8n3
3 + limn→∞
5n + lim
n→∞7
n2√n+ lim
n→∞9n3
=2 + 0 + 0 + 0
3 + 0 + 0 + 0=
2
3.
�H (H®�ÆêÆX) ê��4� September 29, 2013 12 / 34
4��oK$�
��y² limn→∞
xnyn
=A
B©Ø��B > 0©K�3N1§én > N1Ñkyn >
B
2©
é?¿ε > 0§�3N2,N3§én > N2§k|xn − A| < εB
4¶ én > N3§k
|yn − B| < εB2
4(|A|+ 1)©Ké¤kn > max{N1,N2,N3}§k∣∣∣∣xnyn − A
B
∣∣∣∣ = ∣∣∣∣xnB − AynynB
∣∣∣∣ 6 |(xn − A)B|+ |A(B − yn)||ynB|
<
εB4 · B + |A| · εB2
4(|A|+1)
|B2 · B|6
ε
2+ε
2= ε.
·�5w�e4�oK$��~f§
limn→∞
2n3 + 4n√n + 6n + 8
3n3 + 5n2 + 7√n + 9
= limn→∞
2 + 4n√n+ 6
n2+ 8
n3
3 + 5n + 7
n2√n+ 9
n3
=2 + lim
n→∞4
n√n+ lim
n→∞6n2
+ limn→∞
8n3
3 + limn→∞
5n + lim
n→∞7
n2√n+ lim
n→∞9n3
=2 + 0 + 0 + 0
3 + 0 + 0 + 0=
2
3.
�H (H®�ÆêÆX) ê��4� September 29, 2013 12 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§
Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©
@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn|
= |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn|
6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§
é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|
6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|
6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.
Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©
w,−2+14 6 2 sin n+1
4 6 2+14 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n©
limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©
KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n
½n
�{xn}§{yn}§{zn}�ê�§� limn→∞
yn = limn→∞
zn = A§A∈R©XJé¤kn§Ñkyn 6 xn 6 zn©@o
limn→∞
xn = A.
y©Ï�{yn}§{zn}4�Ñ�A§Ké?¿ε > 0§�3N1,N2§én > N1§
|yn − A| < 13 ε¶én > N2§|zn − A| < 1
3 ε©@oXJn > max{N1,N2}§kzn − yn = |zn − yn| = |zn − A+ A− yn| 6 |zn − A|+ |yn − A| < 2
3ε.
¤±�N = max{N1,N2}§é¤kn > N§|xn−A|= |xn−yn+yn−A|6 |xn−yn|+ |yn−A|6 (zn−yn)+ |yn−A|<ε.Y%�n^�¥�“é¤kn”§�±�¤“�3N0§é¤kn > N0”"
k5w��{üA^µ¦ limn→∞
(2 sin n+14
)n©w,−2+1
4 6 2 sin n+14 6 2+1
4 ©
Ïd06∣∣(2 sin n+1
4
)n∣∣6 (34)n© limn→∞
(34
)n= lim
n→∞0=0©KdY%�n§
limn→∞
∣∣(2 sin n+14
)n∣∣ = 0§= limn→∞
(2 sin n+14
)n= 0©
�H (H®�ÆêÆX) ê��4� September 29, 2013 13 / 34
Y%�n�~f
~
y²
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
y©w,n√100n 6 n
√1n + 2n + 3n + · · ·+ 100n 6 n
√100 · 100n,
=
100 6 n√1n + 2n + 3n + · · ·+ 100n 6 100 n
√100.
é?¿ε > 0§�N = [ 1log100(1+ε)
]©Ké?¿n > N§Ñk
| n√100− 1| = 100
1n − 1 6 100
1N+1 − 1 < 100log100(1+ε) − 1 = ε.
Ïdd4��½Â§ limn→∞
n√100 = 1§= lim
n→∞100
n√100 = 100© ,��¡§
w, limn→∞
100 = 100©¤±dY%�n§
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
�H (H®�ÆêÆX) ê��4� September 29, 2013 14 / 34
Y%�n�~f
~
y²
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
y©w,n√100n 6 n
√1n + 2n + 3n + · · ·+ 100n 6 n
√100 · 100n,
=
100 6 n√1n + 2n + 3n + · · ·+ 100n 6 100 n
√100.
é?¿ε > 0§�N = [ 1log100(1+ε)
]©Ké?¿n > N§Ñk
| n√100− 1| = 100
1n − 1 6 100
1N+1 − 1 < 100log100(1+ε) − 1 = ε.
Ïdd4��½Â§ limn→∞
n√100 = 1§= lim
n→∞100
n√100 = 100© ,��¡§
w, limn→∞
100 = 100©¤±dY%�n§
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
�H (H®�ÆêÆX) ê��4� September 29, 2013 14 / 34
Y%�n�~f
~
y²
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
y©w,n√100n 6 n
√1n + 2n + 3n + · · ·+ 100n 6 n
√100 · 100n,
=
100 6 n√1n + 2n + 3n + · · ·+ 100n 6 100 n
√100.
é?¿ε > 0§�N = [ 1log100(1+ε)
]©Ké?¿n > N§Ñk
| n√100− 1| = 100
1n − 1 6 100
1N+1 − 1 < 100log100(1+ε) − 1 = ε.
Ïdd4��½Â§ limn→∞
n√100 = 1§= lim
n→∞100
n√100 = 100© ,��¡§
w, limn→∞
100 = 100©¤±dY%�n§
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
�H (H®�ÆêÆX) ê��4� September 29, 2013 14 / 34
Y%�n�~f
~
y²
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
y©w,n√100n 6 n
√1n + 2n + 3n + · · ·+ 100n 6 n
√100 · 100n,
=
100 6 n√1n + 2n + 3n + · · ·+ 100n 6 100 n
√100.
é?¿ε > 0§�N = [ 1log100(1+ε)
]©
Ké?¿n > N§Ñk
| n√100− 1| = 100
1n − 1 6 100
1N+1 − 1 < 100log100(1+ε) − 1 = ε.
Ïdd4��½Â§ limn→∞
n√100 = 1§= lim
n→∞100
n√100 = 100© ,��¡§
w, limn→∞
100 = 100©¤±dY%�n§
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
�H (H®�ÆêÆX) ê��4� September 29, 2013 14 / 34
Y%�n�~f
~
y²
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
y©w,n√100n 6 n
√1n + 2n + 3n + · · ·+ 100n 6 n
√100 · 100n,
=
100 6 n√1n + 2n + 3n + · · ·+ 100n 6 100 n
√100.
é?¿ε > 0§�N = [ 1log100(1+ε)
]©Ké?¿n > N§Ñk
| n√100− 1| = 100
1n − 1
6 1001
N+1 − 1 < 100log100(1+ε) − 1 = ε.
Ïdd4��½Â§ limn→∞
n√100 = 1§= lim
n→∞100
n√100 = 100© ,��¡§
w, limn→∞
100 = 100©¤±dY%�n§
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
�H (H®�ÆêÆX) ê��4� September 29, 2013 14 / 34
Y%�n�~f
~
y²
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
y©w,n√100n 6 n
√1n + 2n + 3n + · · ·+ 100n 6 n
√100 · 100n,
=
100 6 n√1n + 2n + 3n + · · ·+ 100n 6 100 n
√100.
é?¿ε > 0§�N = [ 1log100(1+ε)
]©Ké?¿n > N§Ñk
| n√100− 1| = 100
1n − 1 6 100
1N+1 − 1
< 100log100(1+ε) − 1 = ε.
Ïdd4��½Â§ limn→∞
n√100 = 1§= lim
n→∞100
n√100 = 100© ,��¡§
w, limn→∞
100 = 100©¤±dY%�n§
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
�H (H®�ÆêÆX) ê��4� September 29, 2013 14 / 34
Y%�n�~f
~
y²
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
y©w,n√100n 6 n
√1n + 2n + 3n + · · ·+ 100n 6 n
√100 · 100n,
=
100 6 n√1n + 2n + 3n + · · ·+ 100n 6 100 n
√100.
é?¿ε > 0§�N = [ 1log100(1+ε)
]©Ké?¿n > N§Ñk
| n√100− 1| = 100
1n − 1 6 100
1N+1 − 1 < 100log100(1+ε) − 1 = ε.
Ïdd4��½Â§ limn→∞
n√100 = 1§= lim
n→∞100
n√100 = 100© ,��¡§
w, limn→∞
100 = 100©¤±dY%�n§
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
�H (H®�ÆêÆX) ê��4� September 29, 2013 14 / 34
Y%�n�~f
~
y²
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
y©w,n√100n 6 n
√1n + 2n + 3n + · · ·+ 100n 6 n
√100 · 100n,
=
100 6 n√1n + 2n + 3n + · · ·+ 100n 6 100 n
√100.
é?¿ε > 0§�N = [ 1log100(1+ε)
]©Ké?¿n > N§Ñk
| n√100− 1| = 100
1n − 1 6 100
1N+1 − 1 < 100log100(1+ε) − 1 = ε.
Ïdd4��½Â§ limn→∞
n√100 = 1§= lim
n→∞100
n√100 = 100©
,��¡§
w, limn→∞
100 = 100©¤±dY%�n§
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
�H (H®�ÆêÆX) ê��4� September 29, 2013 14 / 34
Y%�n�~f
~
y²
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
y©w,n√100n 6 n
√1n + 2n + 3n + · · ·+ 100n 6 n
√100 · 100n,
=
100 6 n√1n + 2n + 3n + · · ·+ 100n 6 100 n
√100.
é?¿ε > 0§�N = [ 1log100(1+ε)
]©Ké?¿n > N§Ñk
| n√100− 1| = 100
1n − 1 6 100
1N+1 − 1 < 100log100(1+ε) − 1 = ε.
Ïdd4��½Â§ limn→∞
n√100 = 1§= lim
n→∞100
n√100 = 100© ,��¡§
w, limn→∞
100 = 100©
¤±dY%�n§
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
�H (H®�ÆêÆX) ê��4� September 29, 2013 14 / 34
Y%�n�~f
~
y²
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
y©w,n√100n 6 n
√1n + 2n + 3n + · · ·+ 100n 6 n
√100 · 100n,
=
100 6 n√1n + 2n + 3n + · · ·+ 100n 6 100 n
√100.
é?¿ε > 0§�N = [ 1log100(1+ε)
]©Ké?¿n > N§Ñk
| n√100− 1| = 100
1n − 1 6 100
1N+1 − 1 < 100log100(1+ε) − 1 = ε.
Ïdd4��½Â§ limn→∞
n√100 = 1§= lim
n→∞100
n√100 = 100© ,��¡§
w, limn→∞
100 = 100©¤±dY%�n§
limn→∞
n√1n + 2n + 3n + · · ·+ 100n = 100.
�H (H®�ÆêÆX) ê��4� September 29, 2013 14 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©q�n> 3�§
( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"�én8B��§�n > 3�§k
n2
2n6 32
23< 2§=n2 < 2n+1§
n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©y3 lim
n→∞2
n√3 = 2 lim
n→∞n√3 = 2©�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©
q�n> 3�§( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"�én8B��§�n > 3�§k
n2
2n6 32
23< 2§=n2 < 2n+1§
n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©y3 lim
n→∞2
n√3 = 2 lim
n→∞n√3 = 2©�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©q�n> 3�§
( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"�én8B��§�n > 3�§k
n2
2n6 32
23< 2§=n2 < 2n+1§
n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©y3 lim
n→∞2
n√3 = 2 lim
n→∞n√3 = 2©�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©q�n> 3�§
( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"
�én8B��§�n > 3�§kn2
2n6 32
23< 2§=n2 < 2n+1§
n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©y3 lim
n→∞2
n√3 = 2 lim
n→∞n√3 = 2©�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©q�n> 3�§
( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"�én8B��§�n > 3�§k
n2
2n6 32
23< 2§
=n2 < 2n+1§n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©y3 lim
n→∞2
n√3 = 2 lim
n→∞n√3 = 2©�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©q�n> 3�§
( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"�én8B��§�n > 3�§k
n2
2n6 32
23< 2§=n2 < 2n+1§
n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©
y3 limn→∞
2n√3 = 2 lim
n→∞n√3 = 2©�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©q�n> 3�§
( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"�én8B��§�n > 3�§k
n2
2n6 32
23< 2§=n2 < 2n+1§
n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©y3 lim
n→∞2
n√3 = 2 lim
n→∞n√3 = 2©
�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©q�n> 3�§
( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"�én8B��§�n > 3�§k
n2
2n6 32
23< 2§=n2 < 2n+1§
n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©y3 lim
n→∞2
n√3 = 2 lim
n→∞n√3 = 2©�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©q�n> 3�§
( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"�én8B��§�n > 3�§k
n2
2n6 32
23< 2§=n2 < 2n+1§
n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©y3 lim
n→∞2
n√3 = 2 lim
n→∞n√3 = 2©�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©q�n> 3�§
( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"�én8B��§�n > 3�§k
n2
2n6 32
23< 2§=n2 < 2n+1§
n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©y3 lim
n→∞2
n√3 = 2 lim
n→∞n√3 = 2©�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©q�n> 3�§
( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"�én8B��§�n > 3�§k
n2
2n6 32
23< 2§=n2 < 2n+1§
n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©y3 lim
n→∞2
n√3 = 2 lim
n→∞n√3 = 2©�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞n2 + n
n2 + n√n= lim
n→∞1 + 1
n
1 + 1√n
= 1
limn→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©q�n> 3�§
( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"�én8B��§�n > 3�§k
n2
2n6 32
23< 2§=n2 < 2n+1§
n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©y3 lim
n→∞2
n√3 = 2 lim
n→∞n√3 = 2©�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
ê�4��~f
~
¦ (1) limn→∞
n√n2 + 2n. lim
n→∞
(1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
).
)©(1) w, n√n2 + 2n > n
√2n =2©q�n> 3�§
( n + 1
n
)26( 3 + 1
3
)2< 2©
=(n + 1)2
2n+16
n2
2n"�én8B��§�n > 3�§k
n2
2n6 32
23< 2§=n2 < 2n+1§
n√n2 + 2n 6 n
√2n+1 + 2n = 2 n
√3©y3 lim
n→∞2
n√3 = 2 lim
n→∞n√3 = 2©�dY
%�n limn→∞
n√n2 + 2n = 2©
(2)w,1
n2+ n√n+ · · ·+ n
n2+ n√n6
1
n2+√n+ · · · + n
n2+ n√n6
1
n2+ · · · + n
n2©
=n(n+1)
2
n2 + n√n6
1
n2 +√n+ · · ·+ n
n2 + n√n6
n(n+1)2
n2©
limn→∞
n(n+1)2
n2 + n√n= lim
n→∞1
2· n2 + n
n2 + n√n=
1
2, lim
n→∞
n(n+1)2
n2= lim
n→∞1
2· n
2 + n
n2=
1
2.
Ïd limn→∞
( 1
n2 +√n+
2
n2 + 2√n+ · · ·+ n
n2 + n√n
)=
1
2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 15 / 34
üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶
ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 16 / 34
üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 16 / 34
üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
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üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
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üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©
-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
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üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"
w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
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üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©
@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 16 / 34
üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 16 / 34
üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"
é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 16 / 34
üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©
Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 16 / 34
üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©
qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 16 / 34
üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©
ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 16 / 34
üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 16 / 34
üNÂñ½n
ê�{xn}¡�üN4Oê�§XJéz�n§Ñkxn+1 > xn¶ê�{xn}¡�üN4~ê�§XJéz�n§Ñkxn+1 6 xn©
½n
�{xn}�üN4Oê�§�{xn}kþ.§=�3M§¦�én§xn 6 M©@o{xn}Âñ§= lim
n→∞xn�3"aq/§XJ{xn}�üN4~�ke.
�ê�§@o limn→∞
xn��3"
y©�y²üN4Oê��Âñ5§üN4~ê��Âñ5�aqy�"
�ê�{xn}üN4O�kþ.M©-8ÜX�dx1, x2, x3, . . . , xn, . . .|¤�8Ü"w,8ÜXkþ.M©@od(.�3½n§8ÜX�3þ(.A©
e¡y²AÒ´ê�{xn}�4�"é?¿ε > 0§dþ(.�½Â§�3,�xN ∈ X§¦�xN > A− ε©Ï�ê�{xn}üN4O§@oé?¿n > N§Ñkxn > xN©qA�X�þ.§�xn 6 A©ù�
|xn − A| = A− xn 6 A− xN < ε.
Ïdd4��½Â§A´ê�{xn}�4�"�H (H®�ÆêÆX) ê��4� September 29, 2013 16 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 = 2§�én> 2§xn =1
2
(xn−1 +
1
xn−1
)©y²
{xn}Âñ§¿¦Ù4�"
y©@oén > 2§w,k
xn =1
2
(xn−1 +
1
xn−1
)>
√xn−1 ·
1
xn−1= 1.
Ïd·��±��
xn =1
2
(xn−1 +
1
xn−1
)6
1
2(xn−1 + xn−1) = xn−1.
={xn}�üN4~ke.1�ê�§ÏdÂñ"�A = limn→∞
xn©@o
A = limn→∞
1
2
(xn−1 +
1
xn−1
)=
1
2
(limn→∞
xn−1 +1
limn→∞
xn−1
)=
1
2
(A+
1
A
).
=A2 − 1 = 0§A = 1£d4��Ò5§Ø�UkA = −1¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 17 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 = 2§�én> 2§xn =1
2
(xn−1 +
1
xn−1
)©y²
{xn}Âñ§¿¦Ù4�"
y©@oén > 2§w,k
xn =1
2
(xn−1 +
1
xn−1
)
>
√xn−1 ·
1
xn−1= 1.
Ïd·��±��
xn =1
2
(xn−1 +
1
xn−1
)6
1
2(xn−1 + xn−1) = xn−1.
={xn}�üN4~ke.1�ê�§ÏdÂñ"�A = limn→∞
xn©@o
A = limn→∞
1
2
(xn−1 +
1
xn−1
)=
1
2
(limn→∞
xn−1 +1
limn→∞
xn−1
)=
1
2
(A+
1
A
).
=A2 − 1 = 0§A = 1£d4��Ò5§Ø�UkA = −1¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 17 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 = 2§�én> 2§xn =1
2
(xn−1 +
1
xn−1
)©y²
{xn}Âñ§¿¦Ù4�"
y©@oén > 2§w,k
xn =1
2
(xn−1 +
1
xn−1
)>
√xn−1 ·
1
xn−1= 1.
Ïd·��±��
xn =1
2
(xn−1 +
1
xn−1
)6
1
2(xn−1 + xn−1) = xn−1.
={xn}�üN4~ke.1�ê�§ÏdÂñ"�A = limn→∞
xn©@o
A = limn→∞
1
2
(xn−1 +
1
xn−1
)=
1
2
(limn→∞
xn−1 +1
limn→∞
xn−1
)=
1
2
(A+
1
A
).
=A2 − 1 = 0§A = 1£d4��Ò5§Ø�UkA = −1¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 17 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 = 2§�én> 2§xn =1
2
(xn−1 +
1
xn−1
)©y²
{xn}Âñ§¿¦Ù4�"
y©@oén > 2§w,k
xn =1
2
(xn−1 +
1
xn−1
)>
√xn−1 ·
1
xn−1= 1.
Ïd·��±��
xn =1
2
(xn−1 +
1
xn−1
)
61
2(xn−1 + xn−1) = xn−1.
={xn}�üN4~ke.1�ê�§ÏdÂñ"�A = limn→∞
xn©@o
A = limn→∞
1
2
(xn−1 +
1
xn−1
)=
1
2
(limn→∞
xn−1 +1
limn→∞
xn−1
)=
1
2
(A+
1
A
).
=A2 − 1 = 0§A = 1£d4��Ò5§Ø�UkA = −1¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 17 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 = 2§�én> 2§xn =1
2
(xn−1 +
1
xn−1
)©y²
{xn}Âñ§¿¦Ù4�"
y©@oén > 2§w,k
xn =1
2
(xn−1 +
1
xn−1
)>
√xn−1 ·
1
xn−1= 1.
Ïd·��±��
xn =1
2
(xn−1 +
1
xn−1
)6
1
2(xn−1 + xn−1) = xn−1.
={xn}�üN4~ke.1�ê�§ÏdÂñ"�A = limn→∞
xn©@o
A = limn→∞
1
2
(xn−1 +
1
xn−1
)=
1
2
(limn→∞
xn−1 +1
limn→∞
xn−1
)=
1
2
(A+
1
A
).
=A2 − 1 = 0§A = 1£d4��Ò5§Ø�UkA = −1¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 17 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 = 2§�én> 2§xn =1
2
(xn−1 +
1
xn−1
)©y²
{xn}Âñ§¿¦Ù4�"
y©@oén > 2§w,k
xn =1
2
(xn−1 +
1
xn−1
)>
√xn−1 ·
1
xn−1= 1.
Ïd·��±��
xn =1
2
(xn−1 +
1
xn−1
)6
1
2(xn−1 + xn−1) = xn−1.
={xn}�üN4~ke.1�ê�§ÏdÂñ"
�A = limn→∞
xn©@o
A = limn→∞
1
2
(xn−1 +
1
xn−1
)=
1
2
(limn→∞
xn−1 +1
limn→∞
xn−1
)=
1
2
(A+
1
A
).
=A2 − 1 = 0§A = 1£d4��Ò5§Ø�UkA = −1¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 17 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 = 2§�én> 2§xn =1
2
(xn−1 +
1
xn−1
)©y²
{xn}Âñ§¿¦Ù4�"
y©@oén > 2§w,k
xn =1
2
(xn−1 +
1
xn−1
)>
√xn−1 ·
1
xn−1= 1.
Ïd·��±��
xn =1
2
(xn−1 +
1
xn−1
)6
1
2(xn−1 + xn−1) = xn−1.
={xn}�üN4~ke.1�ê�§ÏdÂñ"�A = limn→∞
xn©
@o
A = limn→∞
1
2
(xn−1 +
1
xn−1
)=
1
2
(limn→∞
xn−1 +1
limn→∞
xn−1
)=
1
2
(A+
1
A
).
=A2 − 1 = 0§A = 1£d4��Ò5§Ø�UkA = −1¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 17 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 = 2§�én> 2§xn =1
2
(xn−1 +
1
xn−1
)©y²
{xn}Âñ§¿¦Ù4�"
y©@oén > 2§w,k
xn =1
2
(xn−1 +
1
xn−1
)>
√xn−1 ·
1
xn−1= 1.
Ïd·��±��
xn =1
2
(xn−1 +
1
xn−1
)6
1
2(xn−1 + xn−1) = xn−1.
={xn}�üN4~ke.1�ê�§ÏdÂñ"�A = limn→∞
xn©@o
A = limn→∞
1
2
(xn−1 +
1
xn−1
)
=1
2
(limn→∞
xn−1 +1
limn→∞
xn−1
)=
1
2
(A+
1
A
).
=A2 − 1 = 0§A = 1£d4��Ò5§Ø�UkA = −1¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 17 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 = 2§�én> 2§xn =1
2
(xn−1 +
1
xn−1
)©y²
{xn}Âñ§¿¦Ù4�"
y©@oén > 2§w,k
xn =1
2
(xn−1 +
1
xn−1
)>
√xn−1 ·
1
xn−1= 1.
Ïd·��±��
xn =1
2
(xn−1 +
1
xn−1
)6
1
2(xn−1 + xn−1) = xn−1.
={xn}�üN4~ke.1�ê�§ÏdÂñ"�A = limn→∞
xn©@o
A = limn→∞
1
2
(xn−1 +
1
xn−1
)=
1
2
(limn→∞
xn−1 +1
limn→∞
xn−1
)=
1
2
(A+
1
A
).
=A2 − 1 = 0§A = 1£d4��Ò5§Ø�UkA = −1¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 17 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 = 2§�én> 2§xn =1
2
(xn−1 +
1
xn−1
)©y²
{xn}Âñ§¿¦Ù4�"
y©@oén > 2§w,k
xn =1
2
(xn−1 +
1
xn−1
)>
√xn−1 ·
1
xn−1= 1.
Ïd·��±��
xn =1
2
(xn−1 +
1
xn−1
)6
1
2(xn−1 + xn−1) = xn−1.
={xn}�üN4~ke.1�ê�§ÏdÂñ"�A = limn→∞
xn©@o
A = limn→∞
1
2
(xn−1 +
1
xn−1
)=
1
2
(limn→∞
xn−1 +1
limn→∞
xn−1
)=
1
2
(A+
1
A
).
=A2 − 1 = 0§A = 1£d4��Ò5§Ø�UkA = −1¤"�H (H®�ÆêÆX) ê��4� September 29, 2013 17 / 34
ê�4��~f
5¿§þ¡~f¥§3¦4�A�L§§7LÄk�¦{xn}�4��3"
ÄKw��~fµx1 = 3§�én > 2§kxn = x2n−1 − 2©�A = limn→∞
xn©
A = limn→∞
xn = limn→∞
(x2n−1 − 2) =(
limn→∞
xn−1
)2− 2 = A2 − 2.
KA = 2½−1©�x2 = 7, x3 = 47, x4 = 2207, . . .vkþ.§�{xn}uÑ"
ØL3?nùa4íê��4�¯K�§��kò�U�4�k)Ñ5§,�|^Y%�ny²ê��4��(Ò´@��"
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
5¿§þ¡~f¥§3¦4�A�L§§7LÄk�¦{xn}�4��3"ÄKw��~fµx1 = 3§�én > 2§kxn = x2n−1 − 2©
�A = limn→∞
xn©
A = limn→∞
xn = limn→∞
(x2n−1 − 2) =(
limn→∞
xn−1
)2− 2 = A2 − 2.
KA = 2½−1©�x2 = 7, x3 = 47, x4 = 2207, . . .vkþ.§�{xn}uÑ"
ØL3?nùa4íê��4�¯K�§��kò�U�4�k)Ñ5§,�|^Y%�ny²ê��4��(Ò´@��"
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
5¿§þ¡~f¥§3¦4�A�L§§7LÄk�¦{xn}�4��3"ÄKw��~fµx1 = 3§�én > 2§kxn = x2n−1 − 2©�A = lim
n→∞xn©
A = limn→∞
xn = limn→∞
(x2n−1 − 2) =(
limn→∞
xn−1
)2− 2 = A2 − 2.
KA = 2½−1©�x2 = 7, x3 = 47, x4 = 2207, . . .vkþ.§�{xn}uÑ"
ØL3?nùa4íê��4�¯K�§��kò�U�4�k)Ñ5§,�|^Y%�ny²ê��4��(Ò´@��"
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
5¿§þ¡~f¥§3¦4�A�L§§7LÄk�¦{xn}�4��3"ÄKw��~fµx1 = 3§�én > 2§kxn = x2n−1 − 2©�A = lim
n→∞xn©
A = limn→∞
xn
= limn→∞
(x2n−1 − 2) =(
limn→∞
xn−1
)2− 2 = A2 − 2.
KA = 2½−1©�x2 = 7, x3 = 47, x4 = 2207, . . .vkþ.§�{xn}uÑ"
ØL3?nùa4íê��4�¯K�§��kò�U�4�k)Ñ5§,�|^Y%�ny²ê��4��(Ò´@��"
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
5¿§þ¡~f¥§3¦4�A�L§§7LÄk�¦{xn}�4��3"ÄKw��~fµx1 = 3§�én > 2§kxn = x2n−1 − 2©�A = lim
n→∞xn©
A = limn→∞
xn = limn→∞
(x2n−1 − 2)
=(
limn→∞
xn−1
)2− 2 = A2 − 2.
KA = 2½−1©�x2 = 7, x3 = 47, x4 = 2207, . . .vkþ.§�{xn}uÑ"
ØL3?nùa4íê��4�¯K�§��kò�U�4�k)Ñ5§,�|^Y%�ny²ê��4��(Ò´@��"
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
5¿§þ¡~f¥§3¦4�A�L§§7LÄk�¦{xn}�4��3"ÄKw��~fµx1 = 3§�én > 2§kxn = x2n−1 − 2©�A = lim
n→∞xn©
A = limn→∞
xn = limn→∞
(x2n−1 − 2) =(
limn→∞
xn−1
)2− 2
= A2 − 2.
KA = 2½−1©�x2 = 7, x3 = 47, x4 = 2207, . . .vkþ.§�{xn}uÑ"
ØL3?nùa4íê��4�¯K�§��kò�U�4�k)Ñ5§,�|^Y%�ny²ê��4��(Ò´@��"
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
5¿§þ¡~f¥§3¦4�A�L§§7LÄk�¦{xn}�4��3"ÄKw��~fµx1 = 3§�én > 2§kxn = x2n−1 − 2©�A = lim
n→∞xn©
A = limn→∞
xn = limn→∞
(x2n−1 − 2) =(
limn→∞
xn−1
)2− 2 = A2 − 2.
KA = 2½−1©�x2 = 7, x3 = 47, x4 = 2207, . . .vkþ.§�{xn}uÑ"
ØL3?nùa4íê��4�¯K�§��kò�U�4�k)Ñ5§,�|^Y%�ny²ê��4��(Ò´@��"
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
5¿§þ¡~f¥§3¦4�A�L§§7LÄk�¦{xn}�4��3"ÄKw��~fµx1 = 3§�én > 2§kxn = x2n−1 − 2©�A = lim
n→∞xn©
A = limn→∞
xn = limn→∞
(x2n−1 − 2) =(
limn→∞
xn−1
)2− 2 = A2 − 2.
KA = 2½−1©
�x2 = 7, x3 = 47, x4 = 2207, . . .vkþ.§�{xn}uÑ"
ØL3?nùa4íê��4�¯K�§��kò�U�4�k)Ñ5§,�|^Y%�ny²ê��4��(Ò´@��"
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
5¿§þ¡~f¥§3¦4�A�L§§7LÄk�¦{xn}�4��3"ÄKw��~fµx1 = 3§�én > 2§kxn = x2n−1 − 2©�A = lim
n→∞xn©
A = limn→∞
xn = limn→∞
(x2n−1 − 2) =(
limn→∞
xn−1
)2− 2 = A2 − 2.
KA = 2½−1©�x2 = 7, x3 = 47, x4 = 2207, . . .vkþ.§�{xn}uÑ"
ØL3?nùa4íê��4�¯K�§��kò�U�4�k)Ñ5§,�|^Y%�ny²ê��4��(Ò´@��"
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
5¿§þ¡~f¥§3¦4�A�L§§7LÄk�¦{xn}�4��3"ÄKw��~fµx1 = 3§�én > 2§kxn = x2n−1 − 2©�A = lim
n→∞xn©
A = limn→∞
xn = limn→∞
(x2n−1 − 2) =(
limn→∞
xn−1
)2− 2 = A2 − 2.
KA = 2½−1©�x2 = 7, x3 = 47, x4 = 2207, . . .vkþ.§�{xn}uÑ"
ØL3?nùa4íê��4�¯K�§��kò�U�4�k)Ñ5§,�|^Y%�ny²ê��4��(Ò´@��"
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
ØL3?nùa4íê��4�¯K�§��kò�U�4�k)Ñ5§,�|^Y%�ny²ê��4��(Ò´@��"
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©
@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2
= limn→∞
x2n = limn→∞
(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n
= limn→∞
(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ùp�?n==´Ï�8c�vky²f (x) =√x´ëY¼ê§±��±
��¦^
A = limn→∞
xn = limn→∞
√2 + xn−1 =
√2 + lim
n→∞xn−1 =
√2 + A.
Ï
dA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√
2 + xn−1 − 2| =∣∣∣∣ (2 + xn−1)− 4√
2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"
e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√2 + xn−1 − 2| =
∣∣∣∣ (2 + xn−1)− 4√2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√
2 + xn−1 − 2|
=∣∣∣∣ (2 + xn−1)− 4√
2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√
2 + xn−1 − 2| =∣∣∣∣ (2 + xn−1)− 4√
2 + xn−1 + 2
∣∣∣∣
61
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√
2 + xn−1 − 2| =∣∣∣∣ (2 + xn−1)− 4√
2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2|
61
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√
2 + xn−1 − 2| =∣∣∣∣ (2 + xn−1)− 4√
2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
2· 12|xn−2 − 2| =1
4|xn−2 − 2|6 1
8|xn−3 − 2|
6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√
2 + xn−1 − 2| =∣∣∣∣ (2 + xn−1)− 4√
2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√
2 + xn−1 − 2| =∣∣∣∣ (2 + xn−1)− 4√
2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©
dY%�n limn→∞
|xn − 2| = 0§= limn→∞
xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√
2 + xn−1 − 2| =∣∣∣∣ (2 + xn−1)− 4√
2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
5¿�0 6 |xn − 2| 6 |x1−2|2n−1 §d lim
n→∞|x1−2|2n−1 = 0�� lim
n→∞|xn − 2| = 0©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."
k^8B{y²xn 6 2©£xn =√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√
2 + xn−1 − 2| =∣∣∣∣ (2 + xn−1)− 4√
2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
ùp���@´´µk)Ñ�U4�A§2yé,�~ê0 < α < 1§k
|xn − A| 6 α|xn−1 − A| 6 α2|xn−2 − A| 6 · · · 6 αn−1|x1 − A|.ù�d lim
n→∞αn−1|x1 − A| = 0�� lim
n→∞|xn − A| = 0©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."
k^8B{y²xn 6 2©£xn =√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√
2 + xn−1 − 2| =∣∣∣∣ (2 + xn−1)− 4√
2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."
k^8B{y²xn 6 2©£xn =√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√
2 + xn−1 − 2| =∣∣∣∣ (2 + xn−1)− 4√
2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤
Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�ê�{xn}½ÂXeµx1 =√2§�én> 2§xn =
√2 + xn−1©y²{xn}
Âñ§¿¦Ù4�"
y©b�{xn}Âñ§�Ù4��uA©@o
A2 =(
limn→∞
xn
)2= lim
n→∞x2n = lim
n→∞(2 + xn−1) = 2 + A.
ÏdA = 2£��A = −1¤"e¡·��y²{xn}�4��(Ò´2©
|xn − 2| = |√
2 + xn−1 − 2| =∣∣∣∣ (2 + xn−1)− 4√
2 + xn−1 + 2
∣∣∣∣6
1
2|xn−1 − 2| 6 1
4|xn−2 − 2| 6 · · · 6 1
2n−1|x1 − 2|.
limn→∞
12n = 0©dY%�n lim
n→∞|xn − 2| = 0§= lim
n→∞xn = 2©
¯¢þ§3þ¡�~f¥§·���±��y²{xn}üN4Okþ."k^8B{y²xn 6 2©£xn =
√2 + xn−1 6
√2 + 2 = 2©¤Ïd
xn =√xn−1 + 2 >
√xn−1 + xn−1 =
√2 · xn−1 >
√xn−1 · xn−1 = xn−1.
�H (H®�ÆêÆX) ê��4� September 29, 2013 18 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."w,én > 2§k
xn =√
xn−1(3− xn−1) 61
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©ù�én > 3§k
xn =√xn−1 · (3− xn−1) >
√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©düNÂñ½n§{xn}
Âñ"�A = limn→∞
xn©@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."
w,én > 2§k
xn =√
xn−1(3− xn−1) 61
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©ù�én > 3§k
xn =√xn−1 · (3− xn−1) >
√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©düNÂñ½n§{xn}
Âñ"�A = limn→∞
xn©@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."w,én > 2§k
xn =√xn−1(3− xn−1)
61
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©ù�én > 3§k
xn =√xn−1 · (3− xn−1) >
√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©düNÂñ½n§{xn}
Âñ"�A = limn→∞
xn©@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."w,én > 2§k
xn =√xn−1(3− xn−1) 6
1
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©ù�én > 3§k
xn =√xn−1 · (3− xn−1) >
√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©düNÂñ½n§{xn}
Âñ"�A = limn→∞
xn©@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."w,én > 2§k
xn =√xn−1(3− xn−1) 6
1
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©
ù�én > 3§k
xn =√xn−1 · (3− xn−1) >
√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©düNÂñ½n§{xn}
Âñ"�A = limn→∞
xn©@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."w,én > 2§k
xn =√xn−1(3− xn−1) 6
1
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©ù�én > 3§k
xn =√
xn−1 · (3− xn−1)
>√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©düNÂñ½n§{xn}
Âñ"�A = limn→∞
xn©@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."w,én > 2§k
xn =√xn−1(3− xn−1) 6
1
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©ù�én > 3§k
xn =√
xn−1 · (3− xn−1) >√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©
düNÂñ½n§{xn}Âñ"�A = lim
n→∞xn©@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."w,én > 2§k
xn =√xn−1(3− xn−1) 6
1
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©ù�én > 3§k
xn =√
xn−1 · (3− xn−1) >√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©düNÂñ½n§{xn}
Âñ"
�A = limn→∞
xn©@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."w,én > 2§k
xn =√xn−1(3− xn−1) 6
1
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©ù�én > 3§k
xn =√
xn−1 · (3− xn−1) >√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©düNÂñ½n§{xn}
Âñ"�A = limn→∞
xn©
@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."w,én > 2§k
xn =√xn−1(3− xn−1) 6
1
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©ù�én > 3§k
xn =√
xn−1 · (3− xn−1) >√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©düNÂñ½n§{xn}
Âñ"�A = limn→∞
xn©@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."w,én > 2§k
xn =√xn−1(3− xn−1) 6
1
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©ù�én > 3§k
xn =√
xn−1 · (3− xn−1) >√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©düNÂñ½n§{xn}
Âñ"�A = limn→∞
xn©@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©
A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."w,én > 2§k
xn =√xn−1(3− xn−1) 6
1
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©ù�én > 3§k
xn =√
xn−1 · (3− xn−1) >√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©düNÂñ½n§{xn}
Âñ"�A = limn→∞
xn©@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�0< x1< 3§�én> 2§xn =√xn−1(3− xn−1)©¦{xn}�4�"
y©ky²{xn}üNk."w,én > 2§k
xn =√xn−1(3− xn−1) 6
1
2(xn−1 + (3− xn−1)) =
3
2.
Ïdxn 6 3− xn©ù�én > 3§k
xn =√
xn−1 · (3− xn−1) >√xn−1 · xn−1 = xn−1.
=én> 2§x2, x3, . . . , xn, . . .üN4O�kþ.32©düNÂñ½n§{xn}
Âñ"�A = limn→∞
xn©@o
A2 = limn→∞
x2n = limn→∞
xn−1(3− xn−1) = A(3− A).
ù�A =32½A = 0©A = 0´Ø�U�§Ï�én > 2§kxn > x1 > 0©
Ïd limn→∞
xn =32©
�H (H®�ÆêÆX) ê��4� September 29, 2013 19 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©
@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)
= 1 +1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1
= 1 +1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©
e¡5y²{xn}�(Âñu√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣
=∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣
=∣∣∣∣2−
√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣
=∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣2−√2 =√2(√2− 1)
6 (√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2|
6 · · · 6 (√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2|
= (√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©
dY%�n§ limn→∞
|xn −√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�x1=1§�én> 2§xn =1+1
1 + xn−1©y²ê�{xn}Âñ§¿¦4�"
y©�A = limn→∞
xn©@o
A = limn→∞
xn = limn→∞
(1 +
1
1 + xn−1
)= 1 +
1
1 + limn→∞
xn−1= 1 +
1
1 + A.
=(A− 1)(1 + A) = 1§A =√2©e¡5y²{xn}�(Âñu
√2©
|xn −√2| =
∣∣∣∣1 + 1
1 + xn−1−√2∣∣∣∣ = ∣∣∣∣1 + xn−1 + 1−
√2(1 + xn−1)
1 + xn−1
∣∣∣∣=
∣∣∣∣2−√2− (
√2− 1)xn−1
1 + xn−1
∣∣∣∣ = ∣∣∣∣(√2− 1)(
√2− xn−1)
1 + xn−1
∣∣∣∣6 (
√2− 1)|xn−1 −
√2| 6 · · · 6 (
√2− 1)n−1|x1 −
√2| = (
√2− 1)n.
0<√2− 1< 1§� lim
n→∞(√2− 1)n = 0©dY%�n§ lim
n→∞|xn −
√2| = 0§
={xn}�4��√2©
�H (H®�ÆêÆX) ê��4� September 29, 2013 20 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
Ù¢)ù�K|^Y%�n�Ч �üNÂñ½n"XJ��¦^üNÂñ½n�{§5¿�dA = 2− 4
3+A�)��U�4��A = 1§Ï
d·�3y²{xn}üN�§I�©0 < x1 6 1�x1 > 1ü«�¹?Ø"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©
·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©
�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©
ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©
Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1
=(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1
> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O"
ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©
�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©
=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©
Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"
Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"
���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©
K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"
�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
�0< x1< 2§�én> 2§xn =2− 4
3 + xn−1©y²{xn}Âñ§¿¦4�"
y©�0 < x1 6 1©·�k|^8B{y²é¤kn§xn 6 1©�xn−1 6 1©
Kxn = 2− 43+xn−1
6 2− 43+1 = 1©ù�[�kxn 6 1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1> 0.
={xn}kþ.1�üN4O" ex1> 1§Ké¤kn§xn> 1©�xn−1> 1©
Kxn = 2− 43+xn−1
> 2− 43+1 = 1©=okxn�u1©Ïd
xn − xn−1 = 2− 4
3 + xn−1− xn−1 =
(1− xn−1)(2 + xn−1)
3 + xn−1< 0.
={xn}ke.1�üN4~"Ïd��{xn}Âñ"���A = limn→∞
xn©K
A = limn→∞
(2− 4
3 + xn−1
)= 2− 4
3 + A.
Ïd)�A = 1£��A = −2¤"�H (H®�ÆêÆX) ê��4� September 29, 2013 21 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"
K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn
= limn→∞
Fn + Fn−1Fn
= 1 + limn→∞
Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn
= 1 + limn→∞
Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤"
∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣
=∣∣∣∣Fn−1Fn
−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣1−
√5 + 1
2= −
√5− 1
2
=Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣∣∣∣∣1x − 1
y
∣∣∣∣ = |x − y ||x | · |y |
, |x − y | = |x | · |y | ·∣∣∣∣1x − 1
y
∣∣∣∣
6(√
5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣ 6√5− 1
2
∣∣∣∣ FnFn−1
−√5 + 1
2
∣∣∣∣2√5− 1
=
√5 + 1
2
6(√
5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣ 6√5− 1
2
∣∣∣∣ FnFn−1
−√5 + 1
2
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣
6 · · · 6(√
5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣ 6√5− 1
2
∣∣∣∣ FnFn−1
−√5 + 1
2
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.
Ï�12 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣ 6√5− 1
2
∣∣∣∣ FnFn−1
−√5 + 1
2
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©
ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣ 6√5− 1
2
∣∣∣∣ FnFn−1
−√5 + 1
2
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©
={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
ÜÅ@ê£Fibonacci¤ê�{Fn}½ÂXeµF1 = F2 = 1§
�én > 3§Fn = Fn−1 + Fn−2©¦{Fn+1
Fn
}�4�"
y©-A�{Fn+1
Fn
}�4�"K
A = limn→∞
Fn+1
Fn= lim
n→∞Fn + Fn−1
Fn= 1 + lim
n→∞Fn−1Fn
= 1 +1
A.
)��A = 12 (√5 + 1)£��A = 1
2 (1−√5)¤" ∣∣∣∣Fn+1
Fn−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn + Fn−1Fn
−√5 + 1
2
∣∣∣∣ = ∣∣∣∣Fn−1Fn−√5− 1
2
∣∣∣∣=
Fn−1Fn·√5− 1
2·∣∣∣∣ FnFn−1
− 2√5− 1
∣∣∣∣ 6√5− 1
2
∣∣∣∣ FnFn−1
−√5 + 1
2
∣∣∣∣6
(√5− 1
2
)2∣∣∣∣Fn−1Fn−2−√5 + 1
2
∣∣∣∣ 6 · · · 6 (√5− 1
2
)n−1∣∣∣∣F2F1 −√5 + 1
2
∣∣∣∣.Ï�1
2 (√5− 1) < 1§¤± lim
n→∞
(12 (√5− 1)
)n= 0©ÏddY%�n���
limn→∞
∣∣Fn+1
Fn−√5+12
∣∣ = 0©={Fn+1
Fn
}�4��1
2 (√5 + 1)©
�H (H®�ÆêÆX) ê��4� September 29, 2013 22 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
XJ-A = limn→∞
xn§@oU��A = 12 (A+ A)§¤±ù«�{´Ø1�"
¯¢þ§²L{ü�O���§Ò�±uyx2 = 1− 12§x3 = 1− 1
2 + 14§
x4 = 1− 12 + 1
4 −18§��"Ïd�±ßÿxn =
n−1∑k=0
(− 1
2
)k©y©^8B{ØJy²xn =
n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
XJ-A = limn→∞
xn§@oU��A = 12 (A+ A)§¤±ù«�{´Ø1�"
¯¢þ§²L{ü�O���§Ò�±uyx2 = 1− 12§x3 = 1− 1
2 + 14§
x4 = 1− 12 + 1
4 −18§��"Ïd�±ßÿxn =
n−1∑k=0
(− 1
2
)k©
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©
Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1Qº
N´ß� limn→∞
xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
A = 12A+ 1
3A+ 1 =⇒ A = 6
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣
6 12 |xn−1 − 6|+ 1
3 |xn−2 − 6|©�Ò´k|xn − 6| 6 5
6 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��|xn−1− 6|6 5
6 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©
ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©
�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
|xn−1 − 6| 6 max{|xn−2 − 6|, |xn−3 − 6|}=⇒ max{|xn−1 − 6|, |xn−2 − 6|} 6 max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}
6(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©
�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
ê�4��~f
~
�x0 = 0§x1 = 1§�én> 2§xn = 12 (xn−1 + xn−2)©¦{xn}�4�"
y©^8B{ØJy²xn =n−1∑k=0
(− 1
2
)k=
1− (−12)
n
1− (−12)©Ïd
limn→∞
xn = limn→∞
1− (−12)
n
1− (−12)
=2
3
(1− lim
n→∞
(− 1
2
)n)=
2
3.
òþK¥4í'XU�xn = 12xn−1 +
13xn−2 + 1QºN´ß� lim
n→∞xn = 6©
w,|xn − 6| =∣∣12 (xn−1 − 6) + 1
3 (xn−2 − 6)∣∣ 6 1
2 |xn−1 − 6|+ 13 |xn−2 − 6|©
�Ò´k|xn − 6| 6 56 max{|xn−1 − 6|, |xn−2 − 6|}©ù�§·���±��
|xn−1− 6|6 56 max{|xn−2− 6|, |xn−3− 6|}6max{|xn−2− 6|, |xn−3− 6|}©�
max{|xn − 6|, |xn−1 − 6|} 6 5
6max{|xn−2 − 6|, |xn−3 − 6|}
6(5
6
)2max{|xn−4−6|, |xn−5−6|}6
(5
6
)3max{|xn−6−6|, |xn−7−6|}6 · · ·
Ïd{|xn − 6|}Âñu0§={xn}Âñu6©�H (H®�ÆêÆX) ê��4� September 29, 2013 23 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~"
w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©
Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"
�A = limn→∞
an§B = limn→∞
bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©
e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©
kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©
y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©
Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©
dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©
-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"
��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"
£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an©
dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
4«m@½n
½n
�k��4«m[a1, b1], [a2, b2], [a3, b3], . . .§÷v[a1, b1] ⊃ [a2, b2] ⊃ · · ·§=ù4«m�¤��4«m@"q� lim
n→∞(bn − an) = 0§=4«m
��ݪ�u0"K�3��ξ ∈ Ráu¤k4«m§=∞⋂n=1
[an, bn] = {ξ}©
y. Ï�[an, bn] ⊃ [an+1, bn+1]§¤±an 6 an+1§bn > bn+1§={an}üN4O§{bn}üN4~" w,{an}kþ.b1§{bn}ke.a1©Ïd{an}�{bn}þÂñ"�A = lim
n→∞an§B = lim
n→∞bn©e¡y²A = B©kyA 6 B©
Ï�é?¿bm§¤k�anÑ�u�ubm§d4���Ò5§A��an�4��7�u�ubm©y3¤k�bnÑ�u�uA§�B��bn�4��7�u�uA©Ké?¿n§kbn > B > A > an§=B − A 6 bn − an©dY%�n§B − A 6 lim
n→∞(bn − an) = 0§=A = B©-ξ = A§Kξ ∈ [an, bn]
é¤kn¤á"��y��5"£�y¤b��3ξ 6= ξ′§ξ�ξ′Ñáu¤k�[an, bn]§=é¤kn§kan 6 ξ, ξ′6 bn§|ξ′ − ξ|6 bn − an© dY%�n§Ak|ξ′ − ξ| = 0§Ï�bn − an → 0§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 24 / 34
Âñuá�M − N�ó
½Â
�x1, x2, x3, . . .���ê�"eé?¿M > 0§Ñ�3g,êN = N(M)§¦�é¤kn > N§Ñk
(i) xn > M. (ii) xn < −M. (iii) |xn| > M.
K·�©O¡ê�{xn}±(i) +∞§(ii) −∞§(iii) ∞�4�§©OP�
(i) limn→∞
xn = +∞, (ii) limn→∞
xn = −∞, (iii) limn→∞
xn =∞.
5¿�Âñuá�ê�Ø´Âñê�§�´uÑ�"
~X§éq> 1§k limn→∞
logq n=+∞©¯¢þ§é?¿M > 0§�N = [qM ]©
@oé¤kn > N§Ñk
logq n > logq(N + 1) > logq(qM) = M.
Âñuá�kaqY%�n�A^"ean 6 bné¿©��nѤá"K
limn→∞
an = +∞ =⇒ limn→∞
bn = +∞, limn→∞
bn = −∞ =⇒ limn→∞
an = −∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 25 / 34
Âñuá�M − N�ó
½Â
�x1, x2, x3, . . .���ê�"eé?¿M > 0§Ñ�3g,êN = N(M)§¦�é¤kn > N§Ñk
(i) xn > M. (ii) xn < −M. (iii) |xn| > M.
K·�©O¡ê�{xn}±(i) +∞§(ii) −∞§(iii) ∞�4�§©OP�
(i) limn→∞
xn = +∞, (ii) limn→∞
xn = −∞, (iii) limn→∞
xn =∞.
é'�eµ
ε− N�óµ∀ε > 0£?¿�¤§∃N > 0§¦�∀n > N§k|xn − A| < ε©
M − N�óµ∀M > 0£?¿�¤§∃N > 0§¦�∀n > N§kxn > M©
5¿�Âñuá�ê�Ø´Âñê�§�´uÑ�"
~X§éq> 1§k limn→∞
logq n=+∞©¯¢þ§é?¿M > 0§�N = [qM ]©
@oé¤kn > N§Ñk
logq n > logq(N + 1) > logq(qM) = M.
Âñuá�kaqY%�n�A^"ean 6 bné¿©��nѤá"K
limn→∞
an = +∞ =⇒ limn→∞
bn = +∞, limn→∞
bn = −∞ =⇒ limn→∞
an = −∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 25 / 34
Âñuá�M − N�ó
½Â
�x1, x2, x3, . . .���ê�"eé?¿M > 0§Ñ�3g,êN = N(M)§¦�é¤kn > N§Ñk
(i) xn > M. (ii) xn < −M. (iii) |xn| > M.
K·�©O¡ê�{xn}±(i) +∞§(ii) −∞§(iii) ∞�4�§©OP�
(i) limn→∞
xn = +∞, (ii) limn→∞
xn = −∞, (iii) limn→∞
xn =∞.
5¿�Âñuá�ê�Ø´Âñê�§�´uÑ�"
~X§éq> 1§k limn→∞
logq n=+∞©¯¢þ§é?¿M > 0§�N = [qM ]©
@oé¤kn > N§Ñk
logq n > logq(N + 1) > logq(qM) = M.
Âñuá�kaqY%�n�A^"ean 6 bné¿©��nѤá"K
limn→∞
an = +∞ =⇒ limn→∞
bn = +∞, limn→∞
bn = −∞ =⇒ limn→∞
an = −∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 25 / 34
Âñuá�M − N�ó
½Â
�x1, x2, x3, . . .���ê�"eé?¿M > 0§Ñ�3g,êN = N(M)§¦�é¤kn > N§Ñk
(i) xn > M. (ii) xn < −M. (iii) |xn| > M.
K·�©O¡ê�{xn}±(i) +∞§(ii) −∞§(iii) ∞�4�§©OP�
(i) limn→∞
xn = +∞, (ii) limn→∞
xn = −∞, (iii) limn→∞
xn =∞.
5¿�Âñuá�ê�Ø´Âñê�§�´uÑ�"
~X§éq> 1§k limn→∞
logq n=+∞©
¯¢þ§é?¿M > 0§�N = [qM ]©
@oé¤kn > N§Ñk
logq n > logq(N + 1) > logq(qM) = M.
Âñuá�kaqY%�n�A^"ean 6 bné¿©��nѤá"K
limn→∞
an = +∞ =⇒ limn→∞
bn = +∞, limn→∞
bn = −∞ =⇒ limn→∞
an = −∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 25 / 34
Âñuá�M − N�ó
½Â
�x1, x2, x3, . . .���ê�"eé?¿M > 0§Ñ�3g,êN = N(M)§¦�é¤kn > N§Ñk
(i) xn > M. (ii) xn < −M. (iii) |xn| > M.
K·�©O¡ê�{xn}±(i) +∞§(ii) −∞§(iii) ∞�4�§©OP�
(i) limn→∞
xn = +∞, (ii) limn→∞
xn = −∞, (iii) limn→∞
xn =∞.
5¿�Âñuá�ê�Ø´Âñê�§�´uÑ�"
~X§éq> 1§k limn→∞
logq n=+∞©¯¢þ§é?¿M > 0§�N = [qM ]©
@oé¤kn > N§Ñk
logq n > logq(N + 1) > logq(qM) = M.
Âñuá�kaqY%�n�A^"ean 6 bné¿©��nѤá"K
limn→∞
an = +∞ =⇒ limn→∞
bn = +∞, limn→∞
bn = −∞ =⇒ limn→∞
an = −∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 25 / 34
Âñuá�M − N�ó
½Â
�x1, x2, x3, . . .���ê�"eé?¿M > 0§Ñ�3g,êN = N(M)§¦�é¤kn > N§Ñk
(i) xn > M. (ii) xn < −M. (iii) |xn| > M.
K·�©O¡ê�{xn}±(i) +∞§(ii) −∞§(iii) ∞�4�§©OP�
(i) limn→∞
xn = +∞, (ii) limn→∞
xn = −∞, (iii) limn→∞
xn =∞.
5¿�Âñuá�ê�Ø´Âñê�§�´uÑ�"
~X§éq> 1§k limn→∞
logq n=+∞©¯¢þ§é?¿M > 0§�N = [qM ]©
@oé¤kn > N§Ñk
logq n > logq(N + 1)
> logq(qM) = M.
Âñuá�kaqY%�n�A^"ean 6 bné¿©��nѤá"K
limn→∞
an = +∞ =⇒ limn→∞
bn = +∞, limn→∞
bn = −∞ =⇒ limn→∞
an = −∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 25 / 34
Âñuá�M − N�ó
½Â
�x1, x2, x3, . . .���ê�"eé?¿M > 0§Ñ�3g,êN = N(M)§¦�é¤kn > N§Ñk
(i) xn > M. (ii) xn < −M. (iii) |xn| > M.
K·�©O¡ê�{xn}±(i) +∞§(ii) −∞§(iii) ∞�4�§©OP�
(i) limn→∞
xn = +∞, (ii) limn→∞
xn = −∞, (iii) limn→∞
xn =∞.
5¿�Âñuá�ê�Ø´Âñê�§�´uÑ�"
~X§éq> 1§k limn→∞
logq n=+∞©¯¢þ§é?¿M > 0§�N = [qM ]©
@oé¤kn > N§Ñk
logq n > logq(N + 1) > logq(qM) = M.
Âñuá�kaqY%�n�A^"ean 6 bné¿©��nѤá"K
limn→∞
an = +∞ =⇒ limn→∞
bn = +∞, limn→∞
bn = −∞ =⇒ limn→∞
an = −∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 25 / 34
Âñuá�M − N�ó
½Â
�x1, x2, x3, . . .���ê�"eé?¿M > 0§Ñ�3g,êN = N(M)§¦�é¤kn > N§Ñk
(i) xn > M. (ii) xn < −M. (iii) |xn| > M.
K·�©O¡ê�{xn}±(i) +∞§(ii) −∞§(iii) ∞�4�§©OP�
(i) limn→∞
xn = +∞, (ii) limn→∞
xn = −∞, (iii) limn→∞
xn =∞.
5¿�Âñuá�ê�Ø´Âñê�§�´uÑ�"
~X§éq> 1§k limn→∞
logq n=+∞©¯¢þ§é?¿M > 0§�N = [qM ]©
@oé¤kn > N§Ñk
logq n > logq(N + 1) > logq(qM) = M.
Âñuá�kaqY%�n�A^"
ean 6 bné¿©��nѤá"K
limn→∞
an = +∞ =⇒ limn→∞
bn = +∞, limn→∞
bn = −∞ =⇒ limn→∞
an = −∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 25 / 34
Âñuá�M − N�ó
½Â
�x1, x2, x3, . . .���ê�"eé?¿M > 0§Ñ�3g,êN = N(M)§¦�é¤kn > N§Ñk
(i) xn > M. (ii) xn < −M. (iii) |xn| > M.
K·�©O¡ê�{xn}±(i) +∞§(ii) −∞§(iii) ∞�4�§©OP�
(i) limn→∞
xn = +∞, (ii) limn→∞
xn = −∞, (iii) limn→∞
xn =∞.
5¿�Âñuá�ê�Ø´Âñê�§�´uÑ�"
~X§éq> 1§k limn→∞
logq n=+∞©¯¢þ§é?¿M > 0§�N = [qM ]©
@oé¤kn > N§Ñk
logq n > logq(N + 1) > logq(qM) = M.
Âñuá�kaqY%�n�A^"ean 6 bné¿©��nѤá"
K
limn→∞
an = +∞ =⇒ limn→∞
bn = +∞, limn→∞
bn = −∞ =⇒ limn→∞
an = −∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 25 / 34
Âñuá�M − N�ó
½Â
�x1, x2, x3, . . .���ê�"eé?¿M > 0§Ñ�3g,êN = N(M)§¦�é¤kn > N§Ñk
(i) xn > M. (ii) xn < −M. (iii) |xn| > M.
K·�©O¡ê�{xn}±(i) +∞§(ii) −∞§(iii) ∞�4�§©OP�
(i) limn→∞
xn = +∞, (ii) limn→∞
xn = −∞, (iii) limn→∞
xn =∞.
5¿�Âñuá�ê�Ø´Âñê�§�´uÑ�"
~X§éq> 1§k limn→∞
logq n=+∞©¯¢þ§é?¿M > 0§�N = [qM ]©
@oé¤kn > N§Ñk
logq n > logq(N + 1) > logq(qM) = M.
Âñuá�kaqY%�n�A^"ean 6 bné¿©��nѤá"K
limn→∞
an = +∞ =⇒ limn→∞
bn = +∞, limn→∞
bn = −∞ =⇒ limn→∞
an = −∞.�H (H®�ÆêÆX) ê��4� September 29, 2013 25 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞©
y©Ø��2k 6 n < 2k+1§=k = [log2 n]©@ok
1 +1
2+ · · ·+ 1
n> 1 +
1
2+
1
3+ · · ·+ 1
2k
= 1 +1
2+(1
3+
1
4
)+(1
5+
1
6+
1
7+
1
8
)+ · · ·+
(1
2k−1 + 1+
1
2k−1 + 2+ · · ·+ 1
2k
)> 1 +
1
2+(1
4+
1
4
)+(1
8+
1
8+
1
8+
1
8
)+ · · ·+
(1
2k+ · · ·+ 1
2k
)= 1 +
1
2+
1
2+
1
2+ · · ·+ 1
2= 1 +
1
2· k.
Ï� limn→∞
log2 n = +∞§¤±
limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 26 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞©
y©Ø��2k 6 n < 2k+1§=k = [log2 n]©
@ok
1 +1
2+ · · ·+ 1
n> 1 +
1
2+
1
3+ · · ·+ 1
2k
= 1 +1
2+(1
3+
1
4
)+(1
5+
1
6+
1
7+
1
8
)+ · · ·+
(1
2k−1 + 1+
1
2k−1 + 2+ · · ·+ 1
2k
)> 1 +
1
2+(1
4+
1
4
)+(1
8+
1
8+
1
8+
1
8
)+ · · ·+
(1
2k+ · · ·+ 1
2k
)= 1 +
1
2+
1
2+
1
2+ · · ·+ 1
2= 1 +
1
2· k.
Ï� limn→∞
log2 n = +∞§¤±
limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 26 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞©
y©Ø��2k 6 n < 2k+1§=k = [log2 n]©@ok
1 +1
2+ · · ·+ 1
n> 1 +
1
2+
1
3+ · · ·+ 1
2k
= 1 +1
2+(1
3+
1
4
)+(1
5+
1
6+
1
7+
1
8
)+ · · ·+
(1
2k−1 + 1+
1
2k−1 + 2+ · · ·+ 1
2k
)> 1 +
1
2+(1
4+
1
4
)+(1
8+
1
8+
1
8+
1
8
)+ · · ·+
(1
2k+ · · ·+ 1
2k
)= 1 +
1
2+
1
2+
1
2+ · · ·+ 1
2= 1 +
1
2· k.
Ï� limn→∞
log2 n = +∞§¤±
limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 26 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞©
y©Ø��2k 6 n < 2k+1§=k = [log2 n]©@ok
1 +1
2+ · · ·+ 1
n> 1 +
1
2+
1
3+ · · ·+ 1
2k
= 1 +1
2+(1
3+
1
4
)+(1
5+
1
6+
1
7+
1
8
)+ · · ·+
(1
2k−1 + 1+
1
2k−1 + 2+ · · ·+ 1
2k
)
> 1 +1
2+(1
4+
1
4
)+(1
8+
1
8+
1
8+
1
8
)+ · · ·+
(1
2k+ · · ·+ 1
2k
)= 1 +
1
2+
1
2+
1
2+ · · ·+ 1
2= 1 +
1
2· k.
Ï� limn→∞
log2 n = +∞§¤±
limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 26 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞©
y©Ø��2k 6 n < 2k+1§=k = [log2 n]©@ok
1 +1
2+ · · ·+ 1
n> 1 +
1
2+
1
3+ · · ·+ 1
2k
= 1 +1
2+(1
3+
1
4
)+(1
5+
1
6+
1
7+
1
8
)+ · · ·+
(1
2k−1 + 1+
1
2k−1 + 2+ · · ·+ 1
2k
)> 1 +
1
2+(1
4+
1
4
)+(1
8+
1
8+
1
8+
1
8
)+ · · ·+
(1
2k+ · · ·+ 1
2k
)
= 1 +1
2+
1
2+
1
2+ · · ·+ 1
2= 1 +
1
2· k.
Ï� limn→∞
log2 n = +∞§¤±
limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 26 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞©
y©Ø��2k 6 n < 2k+1§=k = [log2 n]©@ok
1 +1
2+ · · ·+ 1
n> 1 +
1
2+
1
3+ · · ·+ 1
2k
= 1 +1
2+(1
3+
1
4
)+(1
5+
1
6+
1
7+
1
8
)+ · · ·+
(1
2k−1 + 1+
1
2k−1 + 2+ · · ·+ 1
2k
)> 1 +
1
2+(1
4+
1
4
)+(1
8+
1
8+
1
8+
1
8
)+ · · ·+
(1
2k+ · · ·+ 1
2k
)= 1 +
1
2+
1
2+
1
2+ · · ·+ 1
2= 1 +
1
2· k.
Ï� limn→∞
log2 n = +∞§¤±
limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 26 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞©
y©Ø��2k 6 n < 2k+1§=k = [log2 n]©@ok
1 +1
2+ · · ·+ 1
n> 1 +
1
2+
1
3+ · · ·+ 1
2k
= 1 +1
2+(1
3+
1
4
)+(1
5+
1
6+
1
7+
1
8
)+ · · ·+
(1
2k−1 + 1+
1
2k−1 + 2+ · · ·+ 1
2k
)> 1 +
1
2+(1
4+
1
4
)+(1
8+
1
8+
1
8+
1
8
)+ · · ·+
(1
2k+ · · ·+ 1
2k
)= 1 +
1
2+
1
2+
1
2+ · · ·+ 1
2= 1 +
1
2· k.
Ï� limn→∞
log2 n = +∞§¤±
limn→∞
(1 +
1
2+
1
3+ · · ·+ 1
n
)= +∞.
�H (H®�ÆêÆX) ê��4� September 29, 2013 26 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
22+
1
32+ · · ·+ 1
n2
)�3"
y©düNÂñ½n§�Iy1+ 122
+ · · · + 1n2kþ.=�"�k=[log2 n]©
1 +1
22+ · · ·+ 1
n26 1 +
1
22+
1
32+ · · ·+ 1
(2k+1 − 1)2
= 1+(1
22+
1
32
)+(1
42+
1
52+
1
62+
1
72
)+ · · · +
(1
(2k)2+ · · · + 1
(2k+1 − 1)2
)6 1+
(1
22+
1
22
)+(1
42+
1
42+
1
42+
1
42
)+ · · · +
(1
(2k)2+ · · · + 1
(2k)2
)= 1+
2
22+
4
42+ · · · + 2k
22k= 1+
1
2+
1
4+ · · · + 1
2k=2− 1
2k< 2.
¢Sþ§þKk�{ü�?n�{"5¿�1
22+
1
32+ · · ·+ 1
n26
1
1 · 2+
1
2 · 3+ · · ·+ 1
(n − 1) · n= 1− 1
n< 1.
�^1�«�{�±y² limn→∞
(1 +
1
2α+ · · ·+ 1
nα
)é¢êα > 1Ñ�3"
�H (H®�ÆêÆX) ê��4� September 29, 2013 27 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
22+
1
32+ · · ·+ 1
n2
)�3"
y©düNÂñ½n§�Iy1+ 122
+ · · · + 1n2kþ.=�"
�k=[log2 n]©
1 +1
22+ · · ·+ 1
n26 1 +
1
22+
1
32+ · · ·+ 1
(2k+1 − 1)2
= 1+(1
22+
1
32
)+(1
42+
1
52+
1
62+
1
72
)+ · · · +
(1
(2k)2+ · · · + 1
(2k+1 − 1)2
)6 1+
(1
22+
1
22
)+(1
42+
1
42+
1
42+
1
42
)+ · · · +
(1
(2k)2+ · · · + 1
(2k)2
)= 1+
2
22+
4
42+ · · · + 2k
22k= 1+
1
2+
1
4+ · · · + 1
2k=2− 1
2k< 2.
¢Sþ§þKk�{ü�?n�{"5¿�1
22+
1
32+ · · ·+ 1
n26
1
1 · 2+
1
2 · 3+ · · ·+ 1
(n − 1) · n= 1− 1
n< 1.
�^1�«�{�±y² limn→∞
(1 +
1
2α+ · · ·+ 1
nα
)é¢êα > 1Ñ�3"
�H (H®�ÆêÆX) ê��4� September 29, 2013 27 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
22+
1
32+ · · ·+ 1
n2
)�3"
y©düNÂñ½n§�Iy1+ 122
+ · · · + 1n2kþ.=�"�k=[log2 n]©
1 +1
22+ · · ·+ 1
n26 1 +
1
22+
1
32+ · · ·+ 1
(2k+1 − 1)2
= 1+(1
22+
1
32
)+(1
42+
1
52+
1
62+
1
72
)+ · · · +
(1
(2k)2+ · · · + 1
(2k+1 − 1)2
)6 1+
(1
22+
1
22
)+(1
42+
1
42+
1
42+
1
42
)+ · · · +
(1
(2k)2+ · · · + 1
(2k)2
)= 1+
2
22+
4
42+ · · · + 2k
22k= 1+
1
2+
1
4+ · · · + 1
2k=2− 1
2k< 2.
¢Sþ§þKk�{ü�?n�{"5¿�1
22+
1
32+ · · ·+ 1
n26
1
1 · 2+
1
2 · 3+ · · ·+ 1
(n − 1) · n= 1− 1
n< 1.
�^1�«�{�±y² limn→∞
(1 +
1
2α+ · · ·+ 1
nα
)é¢êα > 1Ñ�3"
�H (H®�ÆêÆX) ê��4� September 29, 2013 27 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
22+
1
32+ · · ·+ 1
n2
)�3"
y©düNÂñ½n§�Iy1+ 122
+ · · · + 1n2kþ.=�"�k=[log2 n]©
1 +1
22+ · · ·+ 1
n26 1 +
1
22+
1
32+ · · ·+ 1
(2k+1 − 1)2
= 1+(1
22+
1
32
)+(1
42+
1
52+
1
62+
1
72
)+ · · · +
(1
(2k)2+ · · · + 1
(2k+1 − 1)2
)
6 1+(1
22+
1
22
)+(1
42+
1
42+
1
42+
1
42
)+ · · · +
(1
(2k)2+ · · · + 1
(2k)2
)= 1+
2
22+
4
42+ · · · + 2k
22k= 1+
1
2+
1
4+ · · · + 1
2k=2− 1
2k< 2.
¢Sþ§þKk�{ü�?n�{"5¿�1
22+
1
32+ · · ·+ 1
n26
1
1 · 2+
1
2 · 3+ · · ·+ 1
(n − 1) · n= 1− 1
n< 1.
�^1�«�{�±y² limn→∞
(1 +
1
2α+ · · ·+ 1
nα
)é¢êα > 1Ñ�3"
�H (H®�ÆêÆX) ê��4� September 29, 2013 27 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
22+
1
32+ · · ·+ 1
n2
)�3"
y©düNÂñ½n§�Iy1+ 122
+ · · · + 1n2kþ.=�"�k=[log2 n]©
1 +1
22+ · · ·+ 1
n26 1 +
1
22+
1
32+ · · ·+ 1
(2k+1 − 1)2
= 1+(1
22+
1
32
)+(1
42+
1
52+
1
62+
1
72
)+ · · · +
(1
(2k)2+ · · · + 1
(2k+1 − 1)2
)6 1+
(1
22+
1
22
)+(1
42+
1
42+
1
42+
1
42
)+ · · · +
(1
(2k)2+ · · · + 1
(2k)2
)
= 1+2
22+
4
42+ · · · + 2k
22k= 1+
1
2+
1
4+ · · · + 1
2k=2− 1
2k< 2.
¢Sþ§þKk�{ü�?n�{"5¿�1
22+
1
32+ · · ·+ 1
n26
1
1 · 2+
1
2 · 3+ · · ·+ 1
(n − 1) · n= 1− 1
n< 1.
�^1�«�{�±y² limn→∞
(1 +
1
2α+ · · ·+ 1
nα
)é¢êα > 1Ñ�3"
�H (H®�ÆêÆX) ê��4� September 29, 2013 27 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
22+
1
32+ · · ·+ 1
n2
)�3"
y©düNÂñ½n§�Iy1+ 122
+ · · · + 1n2kþ.=�"�k=[log2 n]©
1 +1
22+ · · ·+ 1
n26 1 +
1
22+
1
32+ · · ·+ 1
(2k+1 − 1)2
= 1+(1
22+
1
32
)+(1
42+
1
52+
1
62+
1
72
)+ · · · +
(1
(2k)2+ · · · + 1
(2k+1 − 1)2
)6 1+
(1
22+
1
22
)+(1
42+
1
42+
1
42+
1
42
)+ · · · +
(1
(2k)2+ · · · + 1
(2k)2
)= 1+
2
22+
4
42+ · · · + 2k
22k= 1+
1
2+
1
4+ · · · + 1
2k
=2− 1
2k< 2.
¢Sþ§þKk�{ü�?n�{"5¿�1
22+
1
32+ · · ·+ 1
n26
1
1 · 2+
1
2 · 3+ · · ·+ 1
(n − 1) · n= 1− 1
n< 1.
�^1�«�{�±y² limn→∞
(1 +
1
2α+ · · ·+ 1
nα
)é¢êα > 1Ñ�3"
�H (H®�ÆêÆX) ê��4� September 29, 2013 27 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
22+
1
32+ · · ·+ 1
n2
)�3"
y©düNÂñ½n§�Iy1+ 122
+ · · · + 1n2kþ.=�"�k=[log2 n]©
1 +1
22+ · · ·+ 1
n26 1 +
1
22+
1
32+ · · ·+ 1
(2k+1 − 1)2
= 1+(1
22+
1
32
)+(1
42+
1
52+
1
62+
1
72
)+ · · · +
(1
(2k)2+ · · · + 1
(2k+1 − 1)2
)6 1+
(1
22+
1
22
)+(1
42+
1
42+
1
42+
1
42
)+ · · · +
(1
(2k)2+ · · · + 1
(2k)2
)= 1+
2
22+
4
42+ · · · + 2k
22k= 1+
1
2+
1
4+ · · · + 1
2k=2− 1
2k< 2.
¢Sþ§þKk�{ü�?n�{"5¿�1
22+
1
32+ · · ·+ 1
n26
1
1 · 2+
1
2 · 3+ · · ·+ 1
(n − 1) · n= 1− 1
n< 1.
�^1�«�{�±y² limn→∞
(1 +
1
2α+ · · ·+ 1
nα
)é¢êα > 1Ñ�3"
�H (H®�ÆêÆX) ê��4� September 29, 2013 27 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
22+
1
32+ · · ·+ 1
n2
)�3"
y©düNÂñ½n§�Iy1+ 122
+ · · · + 1n2kþ.=�"�k=[log2 n]©
1 +1
22+ · · ·+ 1
n26 1 +
1
22+
1
32+ · · ·+ 1
(2k+1 − 1)2
= 1+(1
22+
1
32
)+(1
42+
1
52+
1
62+
1
72
)+ · · · +
(1
(2k)2+ · · · + 1
(2k+1 − 1)2
)6 1+
(1
22+
1
22
)+(1
42+
1
42+
1
42+
1
42
)+ · · · +
(1
(2k)2+ · · · + 1
(2k)2
)= 1+
2
22+
4
42+ · · · + 2k
22k= 1+
1
2+
1
4+ · · · + 1
2k=2− 1
2k< 2.
¢Sþ§þKk�{ü�?n�{"
5¿�1
22+
1
32+ · · ·+ 1
n26
1
1 · 2+
1
2 · 3+ · · ·+ 1
(n − 1) · n= 1− 1
n< 1.
�^1�«�{�±y² limn→∞
(1 +
1
2α+ · · ·+ 1
nα
)é¢êα > 1Ñ�3"
�H (H®�ÆêÆX) ê��4� September 29, 2013 27 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
22+
1
32+ · · ·+ 1
n2
)�3"
y©düNÂñ½n§�Iy1+ 122
+ · · · + 1n2kþ.=�"�k=[log2 n]©
1 +1
22+ · · ·+ 1
n26 1 +
1
22+
1
32+ · · ·+ 1
(2k+1 − 1)2
= 1+(1
22+
1
32
)+(1
42+
1
52+
1
62+
1
72
)+ · · · +
(1
(2k)2+ · · · + 1
(2k+1 − 1)2
)6 1+
(1
22+
1
22
)+(1
42+
1
42+
1
42+
1
42
)+ · · · +
(1
(2k)2+ · · · + 1
(2k)2
)= 1+
2
22+
4
42+ · · · + 2k
22k= 1+
1
2+
1
4+ · · · + 1
2k=2− 1
2k< 2.
¢Sþ§þKk�{ü�?n�{"5¿�1
22+
1
32+ · · ·+ 1
n26
1
1 · 2+
1
2 · 3+ · · ·+ 1
(n − 1) · n
= 1− 1
n< 1.
�^1�«�{�±y² limn→∞
(1 +
1
2α+ · · ·+ 1
nα
)é¢êα > 1Ñ�3"
�H (H®�ÆêÆX) ê��4� September 29, 2013 27 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
22+
1
32+ · · ·+ 1
n2
)�3"
y©düNÂñ½n§�Iy1+ 122
+ · · · + 1n2kþ.=�"�k=[log2 n]©
1 +1
22+ · · ·+ 1
n26 1 +
1
22+
1
32+ · · ·+ 1
(2k+1 − 1)2
= 1+(1
22+
1
32
)+(1
42+
1
52+
1
62+
1
72
)+ · · · +
(1
(2k)2+ · · · + 1
(2k+1 − 1)2
)6 1+
(1
22+
1
22
)+(1
42+
1
42+
1
42+
1
42
)+ · · · +
(1
(2k)2+ · · · + 1
(2k)2
)= 1+
2
22+
4
42+ · · · + 2k
22k= 1+
1
2+
1
4+ · · · + 1
2k=2− 1
2k< 2.
¢Sþ§þKk�{ü�?n�{"5¿�1
22+
1
32+ · · ·+ 1
n26
1
1 · 2+
1
2 · 3+ · · ·+ 1
(n − 1) · n= 1− 1
n< 1.
�^1�«�{�±y² limn→∞
(1 +
1
2α+ · · ·+ 1
nα
)é¢êα > 1Ñ�3"
�H (H®�ÆêÆX) ê��4� September 29, 2013 27 / 34
ëY��ê��êÚ
~
y² limn→∞
(1 +
1
22+
1
32+ · · ·+ 1
n2
)�3"
y©düNÂñ½n§�Iy1+ 122
+ · · · + 1n2kþ.=�"�k=[log2 n]©
1 +1
22+ · · ·+ 1
n26 1 +
1
22+
1
32+ · · ·+ 1
(2k+1 − 1)2
= 1+(1
22+
1
32
)+(1
42+
1
52+
1
62+
1
72
)+ · · · +
(1
(2k)2+ · · · + 1
(2k+1 − 1)2
)6 1+
(1
22+
1
22
)+(1
42+
1
42+
1
42+
1
42
)+ · · · +
(1
(2k)2+ · · · + 1
(2k)2
)= 1+
2
22+
4
42+ · · · + 2k
22k= 1+
1
2+
1
4+ · · · + 1
2k=2− 1
2k< 2.
¢Sþ§þKk�{ü�?n�{"5¿�1
22+
1
32+ · · ·+ 1
n26
1
1 · 2+
1
2 · 3+ · · ·+ 1
(n − 1) · n= 1− 1
n< 1.
�^1�«�{�±y² limn→∞
(1 +
1
2α+ · · ·+ 1
nα
)é¢êα > 1Ñ�3"
�H (H®�ÆêÆX) ê��4� September 29, 2013 27 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))
=1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
2 · 3+
1
2 · 3− 1
3 · 4+ · · ·+ 1
(n − 1) · n− 1
n · (n + 1)
)
=1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
2 · 3+
1
2 · 3− 1
3 · 4+ · · ·+ 1
(n − 1) · n− 1
n · (n + 1)
)=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)
=1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
2 · 3+
1
2 · 3− 1
3 · 4+ · · ·+ 1
(n − 1) · n− 1
n · (n + 1)
)=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n
= limn→∞
n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n
=1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a
= limn→∞
(1− a4)(1 + a4) · · · (1 + a2n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a
= limn→∞
1− a2n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
ê�4��~f
~
(1) limn→∞
(1
1 · 2 · 3+ · · · + 1
(n− 1) · n · (n+1)
)(2) lim
n→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)(3) lim
n→∞(1 + a)(1 + a2)(1 + a4)(1 + a8) · · · (1 + a2
n)§ùp|a| < 1©
)©(1) limn→∞
(1
1 · 2 · 3+
1
2 · 3 · 4+ · · ·+ 1
(n − 1) · n · (n + 1)
)=
1
2limn→∞
(1
2
(1
1− 1
3
)+
1
3
(1
2− 1
4
)+ · · ·+ 1
n
(1
n − 1− 1
n + 1
))=
1
2limn→∞
(1
1 · 2− 1
n · (n + 1)
)=
1
4.
(2) limn→∞
(1− 1
22
)(1− 1
32
)· · ·
(1− 1
n2
)= lim
n→∞22 − 1
22· 3
2 − 1
32· · · n
2 − 1
n2
= limn→∞
1 · 32 · 2· 2 · 43 · 3· 3 · 54 · 4· 4 · 65 · 5· · · (n− 2) · n
(n− 1) · (n− 1)· (n− 1) · (n+1)
n · n= lim
n→∞n+1
2n=
1
2.
(3) limn→∞
(1 + a)(1 + a2) · · · (1 + a2n) = lim
n→∞(1− a)(1 + a)(1 + a2) · · · (1 + a2
n)
1− a
= limn→∞
(1− a2)(1 + a2)(1 + a4) · · · (1 + a2n)
1− a= lim
n→∞(1− a4)(1 + a4) · · · (1 + a2
n)
1− a
= · · · = limn→∞
(1− a2n)(1 + a2
n)
1− a= lim
n→∞1− a2
n+1
1− a
|a|<1=====
1
1− a.
�H (H®�ÆêÆX) ê��4� September 29, 2013 28 / 34
���4�
½n
4� limn→∞
(1 +
1
n
)n
�3"
y©ky²{(
1+ 1n
)n}�üN4Oê�"dAÛ²þ-�â²þØ�ª§(
1 +1
n
)n
= 1 ·(1 +
1
n
)n
6(1 + n · (1 + 1
n )
n + 1
)n+1
=(1 +
1
n + 1
)n+1
.
2y²{(
1 + 1n
)n+1}�üN4~ê�§=
{( nn+1
)n+1}�üN4Oê�"(
n
n + 1
)n+1
= 1 ·(
n
n + 1
)n+1
6(1 + (n + 1) · n
n+1
n + 2
)n+2
=(n + 1
n + 2
)n+2
.
Ïd(1 + 1
n
)n<(1 + 1
n
)n+16 4©=
{(1 + 1
n
)n}4Okþ.§Âñ"
·�ò{(
1 + 1n
)n}�4�P�e§=
e = limn→∞
(1 +
1
n
)n
= limn→∞
(1 +
1
n
)n+1
.
�H (H®�ÆêÆX) ê��4� September 29, 2013 29 / 34
���4�
½n
4� limn→∞
(1 +
1
n
)n
�3"
y©ky²{(
1+ 1n
)n}�üN4Oê�"
dAÛ²þ-�â²þØ�ª§(1 +
1
n
)n
= 1 ·(1 +
1
n
)n
6(1 + n · (1 + 1
n )
n + 1
)n+1
=(1 +
1
n + 1
)n+1
.
2y²{(
1 + 1n
)n+1}�üN4~ê�§=
{( nn+1
)n+1}�üN4Oê�"(
n
n + 1
)n+1
= 1 ·(
n
n + 1
)n+1
6(1 + (n + 1) · n
n+1
n + 2
)n+2
=(n + 1
n + 2
)n+2
.
Ïd(1 + 1
n
)n<(1 + 1
n
)n+16 4©=
{(1 + 1
n
)n}4Okþ.§Âñ"
·�ò{(
1 + 1n
)n}�4�P�e§=
e = limn→∞
(1 +
1
n
)n
= limn→∞
(1 +
1
n
)n+1
.
�H (H®�ÆêÆX) ê��4� September 29, 2013 29 / 34
���4�
½n
4� limn→∞
(1 +
1
n
)n
�3"
y©ky²{(
1+ 1n
)n}�üN4Oê�"dAÛ²þ-�â²þØ�ª§(
1 +1
n
)n
= 1 ·(1 +
1
n
)n
6(1 + n · (1 + 1
n )
n + 1
)n+1
=(1 +
1
n + 1
)n+1
.
2y²{(
1 + 1n
)n+1}�üN4~ê�§=
{( nn+1
)n+1}�üN4Oê�"(
n
n + 1
)n+1
= 1 ·(
n
n + 1
)n+1
6(1 + (n + 1) · n
n+1
n + 2
)n+2
=(n + 1
n + 2
)n+2
.
Ïd(1 + 1
n
)n<(1 + 1
n
)n+16 4©=
{(1 + 1
n
)n}4Okþ.§Âñ"
·�ò{(
1 + 1n
)n}�4�P�e§=
e = limn→∞
(1 +
1
n
)n
= limn→∞
(1 +
1
n
)n+1
.
�H (H®�ÆêÆX) ê��4� September 29, 2013 29 / 34
���4�
½n
4� limn→∞
(1 +
1
n
)n
�3"
y©ky²{(
1+ 1n
)n}�üN4Oê�"dAÛ²þ-�â²þØ�ª§(
1 +1
n
)n
= 1 ·(1 +
1
n
)n
6(1 + n · (1 + 1
n )
n + 1
)n+1
=(1 +
1
n + 1
)n+1
.
2y²{(
1 + 1n
)n+1}�üN4~ê�§=
{( nn+1
)n+1}�üN4Oê�"(
n
n + 1
)n+1
= 1 ·(
n
n + 1
)n+1
6(1 + (n + 1) · n
n+1
n + 2
)n+2
=(n + 1
n + 2
)n+2
.
Ïd(1 + 1
n
)n<(1 + 1
n
)n+16 4©=
{(1 + 1
n
)n}4Okþ.§Âñ"
·�ò{(
1 + 1n
)n}�4�P�e§=
e = limn→∞
(1 +
1
n
)n
= limn→∞
(1 +
1
n
)n+1
.
�H (H®�ÆêÆX) ê��4� September 29, 2013 29 / 34
���4�
½n
4� limn→∞
(1 +
1
n
)n
�3"
y©ky²{(
1+ 1n
)n}�üN4Oê�"dAÛ²þ-�â²þØ�ª§(
1 +1
n
)n
= 1 ·(1 +
1
n
)n
6(1 + n · (1 + 1
n )
n + 1
)n+1
=(1 +
1
n + 1
)n+1
.
2y²{(
1 + 1n
)n+1}�üN4~ê�§=
{( nn+1
)n+1}�üN4Oê�"(
n
n + 1
)n+1
= 1 ·(
n
n + 1
)n+1
6(1 + (n + 1) · n
n+1
n + 2
)n+2
=(n + 1
n + 2
)n+2
.
Ïd(1 + 1
n
)n<(1 + 1
n
)n+16 4©=
{(1 + 1
n
)n}4Okþ.§Âñ"
·�ò{(
1 + 1n
)n}�4�P�e§=
e = limn→∞
(1 +
1
n
)n
= limn→∞
(1 +
1
n
)n+1
.
�H (H®�ÆêÆX) ê��4� September 29, 2013 29 / 34
���4�
½n
4� limn→∞
(1 +
1
n
)n
�3"
y©ky²{(
1+ 1n
)n}�üN4Oê�"dAÛ²þ-�â²þØ�ª§(
1 +1
n
)n
= 1 ·(1 +
1
n
)n
6(1 + n · (1 + 1
n )
n + 1
)n+1
=(1 +
1
n + 1
)n+1
.
2y²{(
1 + 1n
)n+1}�üN4~ê�§=
{( nn+1
)n+1}�üN4Oê�"
(n
n + 1
)n+1
= 1 ·(
n
n + 1
)n+1
6(1 + (n + 1) · n
n+1
n + 2
)n+2
=(n + 1
n + 2
)n+2
.
Ïd(1 + 1
n
)n<(1 + 1
n
)n+16 4©=
{(1 + 1
n
)n}4Okþ.§Âñ"
·�ò{(
1 + 1n
)n}�4�P�e§=
e = limn→∞
(1 +
1
n
)n
= limn→∞
(1 +
1
n
)n+1
.
�H (H®�ÆêÆX) ê��4� September 29, 2013 29 / 34
���4�
½n
4� limn→∞
(1 +
1
n
)n
�3"
y©ky²{(
1+ 1n
)n}�üN4Oê�"dAÛ²þ-�â²þØ�ª§(
1 +1
n
)n
= 1 ·(1 +
1
n
)n
6(1 + n · (1 + 1
n )
n + 1
)n+1
=(1 +
1
n + 1
)n+1
.
2y²{(
1 + 1n
)n+1}�üN4~ê�§=
{( nn+1
)n+1}�üN4Oê�"(
n
n + 1
)n+1
= 1 ·(
n
n + 1
)n+1
6(1 + (n + 1) · n
n+1
n + 2
)n+2
=(n + 1
n + 2
)n+2
.
Ïd(1 + 1
n
)n<(1 + 1
n
)n+16 4©=
{(1 + 1
n
)n}4Okþ.§Âñ"
·�ò{(
1 + 1n
)n}�4�P�e§=
e = limn→∞
(1 +
1
n
)n
= limn→∞
(1 +
1
n
)n+1
.
�H (H®�ÆêÆX) ê��4� September 29, 2013 29 / 34
���4�
½n
4� limn→∞
(1 +
1
n
)n
�3"
y©ky²{(
1+ 1n
)n}�üN4Oê�"dAÛ²þ-�â²þØ�ª§(
1 +1
n
)n
= 1 ·(1 +
1
n
)n
6(1 + n · (1 + 1
n )
n + 1
)n+1
=(1 +
1
n + 1
)n+1
.
2y²{(
1 + 1n
)n+1}�üN4~ê�§=
{( nn+1
)n+1}�üN4Oê�"(
n
n + 1
)n+1
= 1 ·(
n
n + 1
)n+1
6(1 + (n + 1) · n
n+1
n + 2
)n+2
=(n + 1
n + 2
)n+2
.
Ïd(1 + 1
n
)n<(1 + 1
n
)n+16 4©=
{(1 + 1
n
)n}4Okþ.§Âñ"
·�ò{(
1 + 1n
)n}�4�P�e§=
e = limn→∞
(1 +
1
n
)n
= limn→∞
(1 +
1
n
)n+1
.
�H (H®�ÆêÆX) ê��4� September 29, 2013 29 / 34
���4�
½n
4� limn→∞
(1 +
1
n
)n
�3"
y©ky²{(
1+ 1n
)n}�üN4Oê�"dAÛ²þ-�â²þØ�ª§(
1 +1
n
)n
= 1 ·(1 +
1
n
)n
6(1 + n · (1 + 1
n )
n + 1
)n+1
=(1 +
1
n + 1
)n+1
.
2y²{(
1 + 1n
)n+1}�üN4~ê�§=
{( nn+1
)n+1}�üN4Oê�"(
n
n + 1
)n+1
= 1 ·(
n
n + 1
)n+1
6(1 + (n + 1) · n
n+1
n + 2
)n+2
=(n + 1
n + 2
)n+2
.
Ïd(1 + 1
n
)n<(1 + 1
n
)n+16 4©=
{(1 + 1
n
)n}4Okþ.§Âñ"
·�ò{(
1 + 1n
)n}�4�P�e§=
e = limn→∞
(1 +
1
n
)n
= limn→∞
(1 +
1
n
)n+1
.
�H (H®�ÆêÆX) ê��4� September 29, 2013 29 / 34
���4�
½n
4� limn→∞
(1 +
1
n
)n
�3"
y©ky²{(
1+ 1n
)n}�üN4Oê�"dAÛ²þ-�â²þØ�ª§(
1 +1
n
)n
= 1 ·(1 +
1
n
)n
6(1 + n · (1 + 1
n )
n + 1
)n+1
=(1 +
1
n + 1
)n+1
.
2y²{(
1 + 1n
)n+1}�üN4~ê�§=
{( nn+1
)n+1}�üN4Oê�"(
n
n + 1
)n+1
= 1 ·(
n
n + 1
)n+1
6(1 + (n + 1) · n
n+1
n + 2
)n+2
=(n + 1
n + 2
)n+2
.
Ïd(1 + 1
n
)n<(1 + 1
n
)n+16 4© ù´Ï�
(1 + 1
1
)1+1= 4
={(
1 + 1n
)n}4Okþ.§Âñ"
·�ò{(
1 + 1n
)n}�4�P�e§=
e = limn→∞
(1 +
1
n
)n
= limn→∞
(1 +
1
n
)n+1
.
�H (H®�ÆêÆX) ê��4� September 29, 2013 29 / 34
���4�
½n
4� limn→∞
(1 +
1
n
)n
�3"
y©ky²{(
1+ 1n
)n}�üN4Oê�"dAÛ²þ-�â²þØ�ª§(
1 +1
n
)n
= 1 ·(1 +
1
n
)n
6(1 + n · (1 + 1
n )
n + 1
)n+1
=(1 +
1
n + 1
)n+1
.
2y²{(
1 + 1n
)n+1}�üN4~ê�§=
{( nn+1
)n+1}�üN4Oê�"(
n
n + 1
)n+1
= 1 ·(
n
n + 1
)n+1
6(1 + (n + 1) · n
n+1
n + 2
)n+2
=(n + 1
n + 2
)n+2
.
Ïd(1 + 1
n
)n<(1 + 1
n
)n+16 4©=
{(1 + 1
n
)n}4Okþ.§Âñ"
·�ò{(
1 + 1n
)n}�4�P�e§=
e = limn→∞
(1 +
1
n
)n
= limn→∞
(1 +
1
n
)n+1
.
�H (H®�ÆêÆX) ê��4� September 29, 2013 29 / 34
���4�
½n
4� limn→∞
(1 +
1
n
)n
�3"
y©ky²{(
1+ 1n
)n}�üN4Oê�"dAÛ²þ-�â²þØ�ª§(
1 +1
n
)n
= 1 ·(1 +
1
n
)n
6(1 + n · (1 + 1
n )
n + 1
)n+1
=(1 +
1
n + 1
)n+1
.
2y²{(
1 + 1n
)n+1}�üN4~ê�§=
{( nn+1
)n+1}�üN4Oê�"(
n
n + 1
)n+1
= 1 ·(
n
n + 1
)n+1
6(1 + (n + 1) · n
n+1
n + 2
)n+2
=(n + 1
n + 2
)n+2
.
Ïd(1 + 1
n
)n<(1 + 1
n
)n+16 4©=
{(1 + 1
n
)n}4Okþ.§Âñ"
·�ò{(
1 + 1n
)n}�4�P�e§=
e = limn→∞
(1 +
1
n
)n
= limn→∞
(1 +
1
n
)n+1
.
�H (H®�ÆêÆX) ê��4� September 29, 2013 29 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§
¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"
�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§
=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©
·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ"
�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n
=1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~"
,��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©
@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n
=1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"
�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
γ~ê
~
4� limn→∞
(1 +
1
2+ · · ·+ 1
n− ln n
)�3§d4�¡�γ~ê"
y©Ï�{(
1 + 1n
)n}üN4O§
{(1 + 1
n
)n+1}üN4~§�§��Âñ4
�Ñ´e§¤±k(1 + 1
n
)n6 e 6
(1 + 1
n
)n+1é¤kn¤á"�g,éê�
�§kln((1 + 1
n
)n)6 ln e 6 ln
((1 + 1
n
)n+1)§=
n ln(1 +
1
n
)6 1 6 (n + 1) ln
(1 +
1
n
),
1
n + 16 ln
(1 +
1
n
)6
1
n.
-xn = 1 + 12 + · · ·+ 1
n − ln n©·�òy²{xn}Âñ" Ï�
xn+1 − xn =1
n + 1− ln(n + 1) + ln n =
1
n + 1− ln
(1 +
1
n
)6 0.
�{xn}üN4~",��¡§-yn = 1 + 12 + · · ·+ 1
n − ln(n + 1)©@o
yn − yn−1 =1
n− ln(n + 1) + ln n =
1
n− ln
(1 +
1
n
)> 0,
={yn}üN4O"�xn> yn > y1=1− ln 2§={xn}ke.§Âñ"�H (H®�ÆêÆX) ê��4� September 29, 2013 30 / 34
fê�
�x1, x2, x3, . . .���ê�§ n1< n2< n3< · · ·´��î�O���ê�"
w,xn1 , xn2 , xn3 , . . .��¤��ê�§ù�ê��P�{xnk}§¡�ê�{xn}���fê�"~X§1, 13 ,
15 ,
17 , . . .§
12 ,
14 ,
16 ,
18 , . . .§1, 14 ,
19 ,
116 , . . .Ñ´ê�1, 12 ,
13 ,
14 , . . .
�fê�"w,��ê����´gC�fê�"
½n
e{xn}ÂñuA§K{xn}�?�fê�{xnk}�ÂñuA§= limk→∞
xnk =A©
��§XJ{xn}�ü�fê�{xnk}�{xmk}§k{xnk}�{xmk
}ÑÂñuA§�{n1, n2, n3, . . .} ∪ {m1,m2,m3, . . .}= {1, 2, 3, . . .}©K{xn}�ÂñuA©
~X§XJ limn→∞
x2n−1 = limn→∞
x2n = A§@o limn→∞
xn = A©
½n
�ê�{xn}kü�fê�{xnk}�{xmk}©b�kü�¢êα, β§α < β§
÷vé¤k�kÑkxnk > β�xmk6 α©@oê�{xn}uÑ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 31 / 34
fê�
�x1, x2, x3, . . .���ê�§ n1< n2< n3< · · ·´��î�O���ê�"w,xn1 , xn2 , xn3 , . . .��¤��ê�§ù�ê��P�{xnk}§¡�ê�{xn}���fê�"
~X§1, 13 ,15 ,
17 , . . .§
12 ,
14 ,
16 ,
18 , . . .§1, 14 ,
19 ,
116 , . . .Ñ´ê�1, 12 ,
13 ,
14 , . . .
�fê�"w,��ê����´gC�fê�"
½n
e{xn}ÂñuA§K{xn}�?�fê�{xnk}�ÂñuA§= limk→∞
xnk =A©
��§XJ{xn}�ü�fê�{xnk}�{xmk}§k{xnk}�{xmk
}ÑÂñuA§�{n1, n2, n3, . . .} ∪ {m1,m2,m3, . . .}= {1, 2, 3, . . .}©K{xn}�ÂñuA©
~X§XJ limn→∞
x2n−1 = limn→∞
x2n = A§@o limn→∞
xn = A©
½n
�ê�{xn}kü�fê�{xnk}�{xmk}©b�kü�¢êα, β§α < β§
÷vé¤k�kÑkxnk > β�xmk6 α©@oê�{xn}uÑ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 31 / 34
fê�
�x1, x2, x3, . . .���ê�§ n1< n2< n3< · · ·´��î�O���ê�"w,xn1 , xn2 , xn3 , . . .��¤��ê�§ù�ê��P�{xnk}§¡�ê�{xn}���fê�"~X§1, 13 ,
15 ,
17 , . . .§
12 ,
14 ,
16 ,
18 , . . .§1, 14 ,
19 ,
116 , . . .Ñ´ê�1, 12 ,
13 ,
14 , . . .
�fê�"
w,��ê����´gC�fê�"
½n
e{xn}ÂñuA§K{xn}�?�fê�{xnk}�ÂñuA§= limk→∞
xnk =A©
��§XJ{xn}�ü�fê�{xnk}�{xmk}§k{xnk}�{xmk
}ÑÂñuA§�{n1, n2, n3, . . .} ∪ {m1,m2,m3, . . .}= {1, 2, 3, . . .}©K{xn}�ÂñuA©
~X§XJ limn→∞
x2n−1 = limn→∞
x2n = A§@o limn→∞
xn = A©
½n
�ê�{xn}kü�fê�{xnk}�{xmk}©b�kü�¢êα, β§α < β§
÷vé¤k�kÑkxnk > β�xmk6 α©@oê�{xn}uÑ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 31 / 34
fê�
�x1, x2, x3, . . .���ê�§ n1< n2< n3< · · ·´��î�O���ê�"w,xn1 , xn2 , xn3 , . . .��¤��ê�§ù�ê��P�{xnk}§¡�ê�{xn}���fê�"~X§1, 13 ,
15 ,
17 , . . .§
12 ,
14 ,
16 ,
18 , . . .§1, 14 ,
19 ,
116 , . . .Ñ´ê�1, 12 ,
13 ,
14 , . . .
�fê�"w,��ê����´gC�fê�"
½n
e{xn}ÂñuA§K{xn}�?�fê�{xnk}�ÂñuA§= limk→∞
xnk =A©
��§XJ{xn}�ü�fê�{xnk}�{xmk}§k{xnk}�{xmk
}ÑÂñuA§�{n1, n2, n3, . . .} ∪ {m1,m2,m3, . . .}= {1, 2, 3, . . .}©K{xn}�ÂñuA©
~X§XJ limn→∞
x2n−1 = limn→∞
x2n = A§@o limn→∞
xn = A©
½n
�ê�{xn}kü�fê�{xnk}�{xmk}©b�kü�¢êα, β§α < β§
÷vé¤k�kÑkxnk > β�xmk6 α©@oê�{xn}uÑ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 31 / 34
fê�
�x1, x2, x3, . . .���ê�§ n1< n2< n3< · · ·´��î�O���ê�"w,xn1 , xn2 , xn3 , . . .��¤��ê�§ù�ê��P�{xnk}§¡�ê�{xn}���fê�"~X§1, 13 ,
15 ,
17 , . . .§
12 ,
14 ,
16 ,
18 , . . .§1, 14 ,
19 ,
116 , . . .Ñ´ê�1, 12 ,
13 ,
14 , . . .
�fê�"w,��ê����´gC�fê�"
½n
e{xn}ÂñuA§K{xn}�?�fê�{xnk}�ÂñuA§= limk→∞
xnk =A©
��§XJ{xn}�ü�fê�{xnk}�{xmk}§k{xnk}�{xmk
}ÑÂñuA§�{n1, n2, n3, . . .} ∪ {m1,m2,m3, . . .}= {1, 2, 3, . . .}©K{xn}�ÂñuA©
~X§XJ limn→∞
x2n−1 = limn→∞
x2n = A§@o limn→∞
xn = A©
½n
�ê�{xn}kü�fê�{xnk}�{xmk}©b�kü�¢êα, β§α < β§
÷vé¤k�kÑkxnk > β�xmk6 α©@oê�{xn}uÑ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 31 / 34
fê�
�x1, x2, x3, . . .���ê�§ n1< n2< n3< · · ·´��î�O���ê�"w,xn1 , xn2 , xn3 , . . .��¤��ê�§ù�ê��P�{xnk}§¡�ê�{xn}���fê�"~X§1, 13 ,
15 ,
17 , . . .§
12 ,
14 ,
16 ,
18 , . . .§1, 14 ,
19 ,
116 , . . .Ñ´ê�1, 12 ,
13 ,
14 , . . .
�fê�"w,��ê����´gC�fê�"
½n
e{xn}ÂñuA§K{xn}�?�fê�{xnk}�ÂñuA§= limk→∞
xnk =A©
��§XJ{xn}�ü�fê�{xnk}�{xmk}§k{xnk}�{xmk
}ÑÂñuA§�{n1, n2, n3, . . .} ∪ {m1,m2,m3, . . .}= {1, 2, 3, . . .}©K{xn}�ÂñuA©
¯¢þ§XJ{xnk}�{xmk}ÑÂñuA©
@o�3K1�K2§�k > K1�§|xnk − A| < ε¶�k > K2�§|xmk
− A| < ε©��n > N = max{nK1 ,mK2}�§7k|xn − A| < ε©
~X§XJ limn→∞
x2n−1 = limn→∞
x2n = A§@o limn→∞
xn = A©
½n
�ê�{xn}kü�fê�{xnk}�{xmk}©b�kü�¢êα, β§α < β§
÷vé¤k�kÑkxnk > β�xmk6 α©@oê�{xn}uÑ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 31 / 34
fê�
�x1, x2, x3, . . .���ê�§ n1< n2< n3< · · ·´��î�O���ê�"w,xn1 , xn2 , xn3 , . . .��¤��ê�§ù�ê��P�{xnk}§¡�ê�{xn}���fê�"~X§1, 13 ,
15 ,
17 , . . .§
12 ,
14 ,
16 ,
18 , . . .§1, 14 ,
19 ,
116 , . . .Ñ´ê�1, 12 ,
13 ,
14 , . . .
�fê�"w,��ê����´gC�fê�"
½n
e{xn}ÂñuA§K{xn}�?�fê�{xnk}�ÂñuA§= limk→∞
xnk =A©
��§XJ{xn}�ü�fê�{xnk}�{xmk}§k{xnk}�{xmk
}ÑÂñuA§�{n1, n2, n3, . . .} ∪ {m1,m2,m3, . . .}= {1, 2, 3, . . .}©K{xn}�ÂñuA©
¯¢þ§XJ{xnk}�{xmk}ÑÂñuA©@o�3K1�K2§�k > K1�§
|xnk − A| < ε¶�k > K2�§|xmk− A| < ε©
��n > N = max{nK1 ,mK2}�§7k|xn − A| < ε©
~X§XJ limn→∞
x2n−1 = limn→∞
x2n = A§@o limn→∞
xn = A©
½n
�ê�{xn}kü�fê�{xnk}�{xmk}©b�kü�¢êα, β§α < β§
÷vé¤k�kÑkxnk > β�xmk6 α©@oê�{xn}uÑ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 31 / 34
fê�
�x1, x2, x3, . . .���ê�§ n1< n2< n3< · · ·´��î�O���ê�"w,xn1 , xn2 , xn3 , . . .��¤��ê�§ù�ê��P�{xnk}§¡�ê�{xn}���fê�"~X§1, 13 ,
15 ,
17 , . . .§
12 ,
14 ,
16 ,
18 , . . .§1, 14 ,
19 ,
116 , . . .Ñ´ê�1, 12 ,
13 ,
14 , . . .
�fê�"w,��ê����´gC�fê�"
½n
e{xn}ÂñuA§K{xn}�?�fê�{xnk}�ÂñuA§= limk→∞
xnk =A©
��§XJ{xn}�ü�fê�{xnk}�{xmk}§k{xnk}�{xmk
}ÑÂñuA§�{n1, n2, n3, . . .} ∪ {m1,m2,m3, . . .}= {1, 2, 3, . . .}©K{xn}�ÂñuA©
¯¢þ§XJ{xnk}�{xmk}ÑÂñuA©@o�3K1�K2§�k > K1�§
|xnk − A| < ε¶�k > K2�§|xmk− A| < ε©��n > N = max{nK1 ,mK2}
�§7k|xn − A| < ε©
~X§XJ limn→∞
x2n−1 = limn→∞
x2n = A§@o limn→∞
xn = A©
½n
�ê�{xn}kü�fê�{xnk}�{xmk}©b�kü�¢êα, β§α < β§
÷vé¤k�kÑkxnk > β�xmk6 α©@oê�{xn}uÑ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 31 / 34
fê�
�x1, x2, x3, . . .���ê�§ n1< n2< n3< · · ·´��î�O���ê�"w,xn1 , xn2 , xn3 , . . .��¤��ê�§ù�ê��P�{xnk}§¡�ê�{xn}���fê�"~X§1, 13 ,
15 ,
17 , . . .§
12 ,
14 ,
16 ,
18 , . . .§1, 14 ,
19 ,
116 , . . .Ñ´ê�1, 12 ,
13 ,
14 , . . .
�fê�"w,��ê����´gC�fê�"
½n
e{xn}ÂñuA§K{xn}�?�fê�{xnk}�ÂñuA§= limk→∞
xnk =A©
��§XJ{xn}�ü�fê�{xnk}�{xmk}§k{xnk}�{xmk
}ÑÂñuA§�{n1, n2, n3, . . .} ∪ {m1,m2,m3, . . .}= {1, 2, 3, . . .}©K{xn}�ÂñuA©
¯¢þ§XJ{xnk}�{xmk}ÑÂñuA©@o�3K1�K2§�k > K1�§
|xnk − A| < ε¶�k > K2�§|xmk− A| < ε©��n > N = max{nK1 ,mK2}
�§7k|xn − A| < ε©
~X§XJ limn→∞
x2n−1 = limn→∞
x2n = A§@o limn→∞
xn = A©
½n
�ê�{xn}kü�fê�{xnk}�{xmk}©b�kü�¢êα, β§α < β§
÷vé¤k�kÑkxnk > β�xmk6 α©@oê�{xn}uÑ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 31 / 34
fê�
�x1, x2, x3, . . .���ê�§ n1< n2< n3< · · ·´��î�O���ê�"w,xn1 , xn2 , xn3 , . . .��¤��ê�§ù�ê��P�{xnk}§¡�ê�{xn}���fê�"~X§1, 13 ,
15 ,
17 , . . .§
12 ,
14 ,
16 ,
18 , . . .§1, 14 ,
19 ,
116 , . . .Ñ´ê�1, 12 ,
13 ,
14 , . . .
�fê�"w,��ê����´gC�fê�"
½n
e{xn}ÂñuA§K{xn}�?�fê�{xnk}�ÂñuA§= limk→∞
xnk =A©
��§XJ{xn}�ü�fê�{xnk}�{xmk}§k{xnk}�{xmk
}ÑÂñuA§�{n1, n2, n3, . . .} ∪ {m1,m2,m3, . . .}= {1, 2, 3, . . .}©K{xn}�ÂñuA©
~X§XJ limn→∞
x2n−1 = limn→∞
x2n = A§@o limn→∞
xn = A©
½n
�ê�{xn}kü�fê�{xnk}�{xmk}©b�kü�¢êα, β§α < β§
÷vé¤k�kÑkxnk > β�xmk6 α©@oê�{xn}uÑ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 31 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©
�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"
-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"
Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©
-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©
K[a1, c1]�[c1, b1]¥7k���¹{xn}¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"
~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©
23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©
�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©
d§����fê�xn1 , xn2 , xn3 , . . .§÷vxnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©
Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©
dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
��5½n
Âñ�ê��½´k.�§�´�L5§w,k.ê��7Âñ§~X§1,−1, 1,−1, . . .©�e¡�½nL²k.�ê�7kÂñ�fê�"
½n
�{xn}�k.ê�§@o�3{xn}���Âñ�fê�{xnk}©
y²òV©Ø��{xn}ke.a§e.b§=x1, x2, x3, . . .Ñ��¹34«m
[a, b]¥"-c = 12 (a+ b)§K«m[a, c]�[c, b]¥7k���¹{xn}¥�Ã
¡õ�"Ø��[a, c]�¹{xn}¥�áõ�§¿3Ù¥?���xn1©-
a1 = a§b1 = c9c1 =12 (a1 + b1)©K[a1, c1]�[c1, b1]¥7k���¹{xn}
¥�áõ�"~X§�[c1, b1]¥�¹Ã¡õ�§K�-a2= c1§b2= b1©
ù�k�±���¹{xn}¥Ã¡õ��f«m[a2, b2]©23[a2, b2]¥?���xn2§��n2 > n1©�gaí§Ò�����4«m@[a1, b1]⊃ [a2, b2]⊃ [a3, b3]⊃ · · ·§¿�ék > 1§
kbk+1− ak+1=12 (bk − ak)©d§����fê�xn1 , xn2 , xn3 , . . .§÷v
xnk ∈ [ak , bk ]©Ï�bk − akª�u0§d4«m@½n§�3���ξáu¤k[ak , bk ]©dY%�n§´�ξÒ´{xnk}�4�"
�H (H®�ÆêÆX) ê��4� September 29, 2013 32 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA
�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"
XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA
�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©
@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA
�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA
�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"
�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA
�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©
AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA
�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©
�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA
�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"
�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA
�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©
éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA
�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©
qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA
�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©
K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA
�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA
�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
�ÜÂñ�OOK
½n
ê�{xn}Âñ�¿�^�´é?¿ε> 0§�3N =N(ε)§��n,m>N§Òk|xn − xm| < ε©
y²òV©7�5´w,�"XJxnª�uA§Ké?¿ε > 0§�3N§¦��n > N�§k|xn − A| < ε
2©@oén,m > N§Òk
|xn − xm| 6 |xn − A|+ |A− xm| 6ε
2+
ε
2= ε.
2y¿©5"�é?¿ε> 0§�3N§én,m>N§k|xn− xm|<ε©AO/§�ε = 1§KkN0§�n > N0�k|xn − xN0+1| < 1©�é¤kn§Ñk
|xn| 6 max{|x1|, |x2|, . . . , |xN0 |, |xN0+1|+ 1},={xn}�k.�"�{xn}kÂñfê�{xnk}©�{xnk}ÂñuA©éε> 0§�3N1§¦��nk > N1Òk|xnk − A| < ε
2©qkN2§��n,m > N2§Ò
k|xn − xm| < ε2©K�n > max{N1,N2}�§?�nk > max{N1,N2}§k|xn − A| 6 |xn − xnk |+ |xnk − A| < ε
2+
ε
2= ε.
Ïd{xn}�ÂñuA�H (H®�ÆêÆX) ê��4� September 29, 2013 33 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"
�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©
=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"
£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"
-c = 12 (a+ b)§K[a, c]�[c , b]¥k��
7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"
Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©
-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©
�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§bk+1 − ak+1 =
12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"
d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©
�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©
�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©
=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34
k�CX½n
éX ⊆ R§ekm«m�I1, I2, I3, . . .£ùpIn = (un, vn)¤§¦�X ⊆∞⋃n=1
In©
K¡{In}�¤��X�mCX"�8ÜX÷v§éX�?¿mCX{In}§o�3Ù¥é�k�õ�m«mIn1 , In2 , . . . , Ink§¦�X ⊆ In1 ∪ In2 ∪ · · · Ink©=X�?¿mCXÑ�3k�fmCX§@o¡X�;8"
½n
4«m[a, b]´;8"
y²òV©�{In}´[a, b]���mCX"£�y¤�[a, b]7L�{In}¥�Ã�õ�m«mâUCX4"-c = 1
2 (a+ b)§K[a, c]�[c , b]¥k��7L�{In}¥Ã�õ�m«mâUX4"Ø��[a, c]´ù��§-a1 = a§
b1= c©-c1=12 (a1 + b1)©�gaí�4«m@�[a1, b1]⊃ [a2, b2]⊃ · · ·§
bk+1 − ak+1 =12 (bk − ak)§�[ak , bk ]7L�{In}¥�Ã�õ�m«mâU
X4"d4«m@½n§k���ξáu¤k[ak , bk ]©�ξ7áuI1, I2, . . .¥�,�m«mIn = (un, vn)©�k¿©��§Òk[ak , bk ]⊂ (un, vn)©=��In��m«mÒ�±r[ak , bk ]X4§gñ"
�H (H®�ÆêÆX) ê��4� September 29, 2013 34 / 34