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Page 1: ก 1 ˘ˇˆ˙ ก˛ - WordPress.com...ก 1 ˘ˇˆ˙ ก˛ ˚˜ ˛ !ก ก ก 1. ˛ # ก ˘. 6.1 2. ˜ % ˜ %&’ ก ( 6.1 .6/1-6 3. *˙ ก & ˙+ 3.1 ˜ ! "# - $ % & ˜ ! - ’(

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2. ����%�����%�&' �ก���(��� � 6.1 �.6/1-6

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2. *���* �1�6 2.1 *���ก�������� 1) �$��%���&����� �! 2) �'(!!ก��ก��%� ��� �! 3) ��� �! �&�)*" 4) ��� �! �&��)*" 2.2 �ก7� / ก�����ก�� 1) ��ก��ก��������ก�,4� 2) ��ก��ก��.8=�.������)*"B�."�# 3) ��ก��ก��ก3(F/%� 2.3 �ก7�ก��1�� 1) ��ก��ก���* (��$�� 2) ��ก��ก���* $* ����%# 3) ��ก��ก���* (���,ก"#7�3

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1) -����6�กก�����ก*6ก�����= 1.1 ก 2) -����6�กก�����ก*6ก���ก�,4� 3) -����6�กก�����ก*6ก���!'�)�ก�� G������ �8=����� �!��* "4��HI

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���ก��� �!��=��� �� (A� P" {1, 2, 3, S, n} $4���� �!6��ก� (finite sequence) �� ���ก��� �!��=��� �� (A� P" {1, 2, 3, S, n, S} $4� ��� �!����"# (infinite sequence)

��'.�,(6�ก!��*��� *=� "*�$4� ��� �!� ���=$H 2(!4���ก (A� 2 ��* �8� 1. ��� �!6��ก� %���DC���� �!��=��� �� (A� P"&��6���$� "@�!$ก n 6���$��ก �8� P" {1, 2, 3, S, n} �� ��6#�8� P" {f(1), f(2), f(3), S, f(n)}

Page 5: ก 1 ˘ˇˆ˙ ก˛ - WordPress.com...ก 1 ˘ˇˆ˙ ก˛ ˚˜ ˛ !ก ก ก 1. ˛ # ก ˘. 6.1 2. ˜ % ˜ %&’ ก ( 6.1 .6/1-6 3. *˙ ก & ˙+ 3.1 ˜ ! "# - $ % & ˜ ! - ’(

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2. ��� �!����"# %���DC���� �!��=��� �� (A� P"&��6���$� "@�!$ก�8� P" {1, 2, 3, S, n, S} %�8� P"&��6���$�:�����"* �� ��6#�8� P" {f(1), f(2), f(3), S, f(n), S}

5.2 (�%�*��

ก�+ก���ก�����ก��*�� <=กก��1�����

1. ��'�:*!��"��%�$&3���= 1.1.2 �'(!!ก��ก��%� ��� �!!!"4��H 6�ก��1���� .�� "�$��4��"4�2(��1 *=� "*� �������� 6�%� 5 6�#�ก&����� �! {an} �8=� an ���4� ��"4�2(��1

an = 2n2 − 3n

an = 2n − 1

��> � 1. an = 2n2 − 3n

a1 = 2(1)2 − 3(1) = 2 − 3 = −1

a2 = 2(2)2 − 3(2) = 8 − 6 = 2

a3 = 2(3)2 − 3(3) = 18 − 9 = 9

a4 = 2(4)2 − 3(4) = 32 − 12 = 20

a5 = 2(5)2 − 3(5) = 50 − 15 = 35

2. an = 2n − 1

a1 = 21 − 1 = 2 − 1 = 1

a2 = 22 − 1 = 4 − 1 = 3

a3 = 23 − 1 = 8 − 1 = 7

a4 = 24 − 1 = 16 − 1 = 15

a5 = 25 − 1 = 32 − 1 = 31

�������� 6�%�6�#��=$2(&����� �!PC=��� 5 6�#�ก ��"4�2(��1 1) 1 , 2

1 , 31 , 4

1 , 51 , S, an , S

2) 21 , 3

2 , 43 , 5

4 , S, an , S

��> � 1. a1 = 1

a2 = 21

a3 = 31

a4 = 41

a5 = 51

����1� an = n1

��ก��ก���* $* ����%# ��ก��ก���* (���,ก"#7�3

Page 6: ก 1 ˘ˇˆ˙ ก˛ - WordPress.com...ก 1 ˘ˇˆ˙ ก˛ ˚˜ ˛ !ก ก ก 1. ˛ # ก ˘. 6.1 2. ˜ % ˜ %&’ ก ( 6.1 .6/1-6 3. *˙ ก & ˙+ 3.1 ˜ ! "# - $ % & ˜ ! - ’(

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2. a1 = 21 = 11

1+

a2 = 32 = 12

2+

a3 = 43 = 13

3+

a4 = 54 = 14

4+

����1� an = 1nn+

2. ��'7%3��ก �������ก*6ก�����= 1.1 ก &3���= 1 ���7� $����=ก��%� 6�ก��1� a�����"�!�4$�ก��

3. ��'�:*!��!��*���&����� �! �&�)*"

� ����� ��� �! �&�)*" �8���� �!PC=���-�"4����=2 36�กก�����6�#��= n+1 �! 3$�6�#��= n (A��4���"�$��= �4�ก�� .��%��!�,ก6���$� "@�!$ก n �� ���ก �4���"�$��= (A�-�"4����1$4� -�"4���4$�

�3$�:*!����=��&��.'"� an = a1+ (n − 1)d ����1

D3�7%3 d ��-�"4���4$���%$4��6�#��= n + 1 ก�!6�#��= n

6�2 3 d = an+1− an �8=� n∈I+

D3�7%3 a1 , a2 , a3 , a4 , ..., an , an+1 , ... (A���� �! �&�)*"�3$

d = a2− a1 = a3− a2 = a4− a3 = ... = an+1 − an = ...

a2 = a1+ d

a3 = a2+ d = (a1+ d) + d = a1+ 2d

a4 = a3+ d = (a1+ 2d) + d = a1+ 3d

M an = an−1+ d = (a1+ (n − 2)d) + d = a1+ (n − 1)d

����1� �8=�ก��%� 7%3 a1 (A�6�#�ก&����� �! �&�)*" �� d (A�

-�"4���4$� 6�#��= n &����� �! �&�)*" �8�

an = a1+ (n − 1)d

��ก��ก���* (��$��

��ก��ก���* $* ����%#

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4. ��'7%3��ก ����BCก��"�$��4����= 1 7�%���.8� ���� ���ก"�$��4�� *=� "*� ����1 1) 6�%�6�#��= n &����� �!"4�2(��1

(1) 9, 13, 17, 21, ... (2) 1, 1.5, 2, 2.5, ... ��> � (1) 9, 13, 17, 21, S

a1 = 9, d = 13 − 9 = 17 − 13 = 21 − 17 = 4

6�#��= n &����� �! �&�)*" �8� an = a1+ (n − 1)d

∴ an = 9 + (n − 1)4

= 9 + 4n − 4 = 5 + 4n

(2) 1, 1.5, 2, 2.5, ... a1 = 1, d = 1.5 − 1 = 2 − 1.5 = 2.5 − 2 = 0.5

6�ก.'"� an = a1+ (n − 1)d

∴ an = 1 + (n − 1) 0.5

= 1 + 0.5n − 0.5 = 0.5n + 0.5

2) ก��%� ��� �! �&�)*" 24, 19, 14, 9, ..., −46 6�%�$4���� �!��1��ก�=6�# ��6�#��= 12 (A� �4�2� ��> � 6�ก��� �! �&�)*" 24, 19, 14, 9, ..., −46

a1 = 24

d = 19 − 24 = 14 − 19 = 9 − 14 = S = −5 �� an = −46

6�ก.'"� an = a1+ (n − 1)d

���4� −46 = 24 + (n − 1) (−5) −46 = 24 − 5n + 5

5n = 29 + 46 5n = 75

n = 575 = 15

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Page 8: ก 1 ˘ˇˆ˙ ก˛ - WordPress.com...ก 1 ˘ˇˆ˙ ก˛ ˚˜ ˛ !ก ก ก 1. ˛ # ก ˘. 6.1 2. ˜ % ˜ %&’ ก ( 6.1 .6/1-6 3. *˙ ก & ˙+ 3.1 ˜ ! "# - $ % & ˜ ! - ’(

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ก�+ก���ก�����ก��*�� <=กก��1�����

6�#��= 12 �8� a12 = a1 + (12 − 1)d

= 24 + 11(−5) = 24 − 55 = − 31

����1� ��� �! 24, 19, 14, 9, ..., −46 �� 15 6�# ��6�#��= 12 �4�ก�! −31

5. ��'�:*!��!��*���&����� �! �&��)*"

� ����� ��� �! �&��)*" �8���� �!PC=�����"��.4$�&��6�#��= n+1 "4�6�#��= n (A��4���"�$��= �4�ก�� .��%��!�,ก6���$� "@�!$ก n �� ���ก�4���"�$��= (A���"��.4$���1$4� ��"��.4$��4$�

�3$�:*!����=��&��.'"� an = a1rn−1 ����1

D3�7%3 r ����"��.4$��4$���%$4��6�#��= n+1 ก�!6�#��= n

6�2 3 r = n

1na

a + %�8� ran = an+1 �8=� n∈I+

r = 12

aa =

23

aa =

34

aa = S =

n1n

aa + = S

a2 = a1r

a3 = a2r = (a1r)r = a1r2 = a1r3−1

a4 = a3r = (a1r2)r = a1r3 = a1r4−1

M

an = a n−1r = (a1r n−2)r = a1rn−1

����1� �8=�ก��%� 7%3 a1 (A�6�#�ก&����� �! �&��)*" �� r (A�

��"��.4$��4$� 6�#��= n &����� �! �&��)*" �8�

an = a1rn−1

6. ��'7%3��ก ����BCก��"�$��4����= 2 7�%���.8� ���� �3$�ก"�$��4�� *=� "*� ����1

1) 6�%�6�#��= n &����� �!"4�2(��1 12, 4, 3

4 , 94 , ...

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ก�+ก���ก�����ก��*�� <=กก��1�����

��> � a1 = 12, r = 4 ÷12 = 34 ÷ 4 = 9

4 ÷ 34 = S = 3

1

6�ก.'"� an = a1rn−1

= 12 ( ) 1n31 −

2) ��� �! �&��)*"��� �!%�C=� ��6�#��= 3 ��กก$4�6�#��= 2 ��'4 149 ��6�#��= 2 ��กก$4�

6�#�ก��'4 73 6�%�6�#��= n &����� �!��1

��> � 6�ก�6��# a3− a2 = 149

����1� a1r2 − a1r = 149

a1r(r − 1) = 149 SSS.(1)

�� a2− a1 = 73

a1r − a1 = 73

a1(r − 1) = 73 SSS.(2)

(1) ÷ (2); 1)(ra1)r(ra

11

−− = 14

9 ÷ 73

r = 149 × 3

7 = 23

���4� r 7� (2); a1 ( )123 − = 7

3

21 a1 = 7

3

a1 = 76

6�#��= n; an = a1rn−1

����1� an = ( ) 1n23

76 −

7. ��'�:*!�� �8=����� �!g��#���*ก *=� "*�6�ก%���.8� ���� ����1 ��� �!g��#���*ก %���DC� ��� �!��=��6�#"4��6�# (A�.4$�ก��!&��6�#7�

��� �! �&�)*" D3� a1 , a2 , a3 , S, an , S (A���� �! �&�)*" �3$ , , , S, , S

(A���� �!g��#���*ก � ���= ai ≠ 0, i = 1, 2, 3, S

�������� 31 , 5

1 , 71 , S, 12n

1+ , S (A���� �!g��#���*ก%�8�2�4 D3� (A� 6�%�6�#��=

15 ��> � �8=��6�ก 3, 5, 7, ..., 2n+1, S (A���� �! �&�)*"

����1� 31 , 5

1 , 71 , ..., 12n

1+ , S (A���� �!g��#���*ก

�� �8=��6�ก an = 2n+1

a15 = 2(15) + 1 = 31

��ก��ก���* (��$��

1 a1

1 a2

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2

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1. lim an = + ∞ D3� a > 1

2. lim an = 1 D3� a = 1

3. lim an = 0 D3� −1 < a < 1

4. lim an 2�4���*�*" D3� a ≤ −1

7. ��'��� .��"�$��4��"4�2(��1 *=� "*�6�ก7�%���.8� ����

1) ก��%� 7%3 an = 1n2n+

�� bn = 2n1n+ 6�%��*�*"&�� an, bn, anbn ��

nn

ba

��> � lim an = lim ( )1n2n+

= lim

+ n11

2

= 012+

= 2 lim bn = lim 2n

1n+

= lim 2n11+

= 201+

= 21

��ก��ก���* (���,ก"#7�3

n → ∞

n → ∞

n → ∞

n → ∞

n → ∞ n → ∞

n → ∞

n → ∞

n → ∞

n → ∞

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20

ก�+ก���ก�����ก��*�� <=กก��1�����

lim an bn = lim an lim bn

= (2) ( )21

= 1

lim n

nba

= nn

nnblimalim

∞→

∞→

= 212

= 4

2) 6�%��*�*"&����� �!PC=���6�#��= n �4�ก�! 6n8n

3)2)(4n1)(3n(2n3 −

−−−+

��> � an = 6n8n

3)2)(4n1)(3n(2n3 −

−−−+

= ( )( )( )

33n

6n8nn 34nn 23nn 12n

−−−

+

an = ( )( )( )

32 n6

n18

n34n23n12−

−−−

+

lim an = ( ) ( ) ( )

3n2nn

nnn

n6limn

1lim8limn34limn23limn12lim

∞→∞→∞→

∞→∞→∞→

−−−

+

= 0080)0)(40)(3(2

−−−−

+

= 824

= 3

3) 6�%��*�*"&����� �!PC=���6�#��= n �4�ก�! 1nnn 22 ++ −

��> � an = 1nnn 22 ++ −

= 1nnn

)1nnn)(1nnn(22

2222

+++

+++++ −

= 1nnn

1nnn22

22

+++

+ −−

n → ∞

n → ∞ n → ∞ n → ∞

n → ∞

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ก�+ก���ก�����ก��*�� <=กก��1�����

= 1nnn

1n22 +++

an = 2n

11n11n11

+++

lim an = 0101

01+++

= 111+

= 21

8. ��'7%3��ก ����!4�ก�,4����ก*6ก�����= 1.1 & ก�,4��� 3 &3� � �$*:�6�!a��ก ���7� $����=ก��%� �3$������"�!����*(����4$�ก��

5.3 (�%�*��&

��'����ก �����4$�ก��.�,("��(�� @�"4�2(��1 1) ��� �!����"#��=���*�*" ���ก$4� ��� �!�'4 &3�

��� �!����"#��=2�4���*�*" ���ก$4� ��� �!�'4��ก 2) ก��*6��)�$4���� �!7 �'4 &3� ��� �!7 �'4��ก���2 3� � - *6��)�6�กก��O&����� �! - 7�3���r�!� ก�=�$ก�!�*�*"

6. *���ก�������� / �����ก��������

6.1 *���ก�������� - %���.8� ���� .�*��)*"B�."�# *=� "*� �.6 ��� ������= 2 - %���.8� ����.���ก�� �����'3 *=� "*� �)*"B�."�# �4� 2 ก�,4�.���ก�� �����'3�)*"B�."�# ��1���:��BCก�� (l��= 6 � �.D�!��.4� .�*�ก��.��$*���B�."�#�� �������� (..$�.) ก����$�BCก��:*ก��

6.2 �����ก�������� 1) %3��.�, ��� ���� 2) %3��.�, ก�,4�.���ก�� �����'3�)*"B�."�#

n → ∞

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7. ก�+ก���*�����

7.1 ก�+ก���*��*���ก��1�������1����� (�%�������(����� 7%3��ก ����!4�ก�,4�BCก��$*:�ก��*.'6�#���r�!� ก�=�$ก�!�*�*"&����� �! (�%���1����� 7%3��ก ����$* ����%#��=����&�1�"��7�ก��*.'6�# ��. �ก��*.'6�#� ��� ���

(�%�*��& 7%3��ก ���� &������r�!� ก�=�$ก�!�*�*"��$*:�ก��*.'6�# 6� ��� (A�������.4���'

(�%�&����ก��.�� ��'����ก �����4$�ก����*(���$*:�ก��*.'6�#���r�!� ก�=�$ก�!�*�*""4��&3�

7.2 ก�+ก������!�ก�� -

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8. ��� :ก����ก��*�� (!���Cก a��(�� @���=��&3��'�.��.� �B�� 6�)

&���E�ก����� :ก +����� +�� �1��&���&���

1. ก��6� ก*6ก���ก�� �����'3

2. ก��7�3.8=�ก�� �����'3

3. ก��(�� �*�-�ก�� �����'3

4. ก��!���,-�ก�� �����'3&����ก ����

��� :ก0������ SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS.S..SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS..SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS..SS

���8=�SSSSSSSSSSSSS..-'3.��

��� :ก1����E�(��2�����+*���2�ก��+��ก�������� SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS.S..SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS.SS.SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS.SS..SS..SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS.SSS

���8=�................................................................

"��%�4�...........................................................

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���*��ก�ก�����1 �F�����ก������ก�+ก���������%�

�������%� ............... ����� ........................................................... ��� � ............ ���� ........................... 0.G. ..................

1 ��%�+� ��'-'3.��(�� �*���ก ����� �7�3$*:�.�� ก"7�&)� �� �*�ก��.�� �3$7%3�� �!���� ����1 3 �8=�(9*!�"*!4��H 2 �8=�(9*!�"*!�����1� 1 �8=�2�4(9*!�"*%�8�(9*!�"*�3����ก

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10-12 7-9 4-6

3 3 3 3 12 � 0�.�� &���&���

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����1!��G�*��� ����� ........................................................... ��� � ............ ���� ........................... 0.G. ....................

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5 5 5 10 25 � 0�.�� &���&���

ก!;�ก��.��1���� 5 �8=��"*ก���� 4��� 6� ���ก (A�!!��4��7%3ก4-'3�8=� 4 �8=��"*ก���� 4� � 3 �8=��"*ก��� ���! �4�ก����=$2( (A�2("����=ก��%� 2 �8=��"*ก���2�4�4��� 4���"=��ก$4���"������=$2( 1 �8=��"*ก���2�4 %���.� 2�4�4��. ���ก%�8�7%3�$���4$��8�

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26

�2�ก��+��ก�������� � 1/3

����� ���ก��������

��� 4 ����4�� 1. &5�����ก��������

1.1 2�ก�������� %�-�!$ก&����,ก������"#2 3������$���'3 �8=����� �!����,ก��2(7�3ก3(F/%�2 3 1.2 +��&��*�1�ก�������� %�-�!$ก&����,ก������"#2 3������$���'3 �8=����� �!����,ก��2(7�3ก3(F/%�2 3

2. *���* �1�6 2.1 *���ก�������� ��,ก������"# 2.2 �ก7� / ก�����ก�� 1) ��ก��ก��������ก�,4� 2) ��ก��ก��.8=�.������)*"B�."�# 3) ��ก��ก��ก3(F/%� 2.3 �ก7�ก��1�� 1) ��ก��ก���* (��$�� 2) ��ก��ก���* (���,ก"#7�3 3) ��ก��ก���* $* ����%#

3. �������ก�������� 3.1 2���� / ��%����

1) -����6�กก�����ก*6ก�����= 1.2 ก 2) -����6�กก�����ก*6ก���ก�,4� 3) -����6�กก�����ก*6ก���!'�)�ก�� G�6��#�������ก�� ก�=�$ก�!ก��%�-�!$ก&����,ก������"#I 3.2 ก�����ก��(�%����ก��&8�������� 1) 6� ก�,4� ก�,4��� 5-6 �� 2) �8�ก%�$%�3�ก�,4� 3) %�$%�3�ก�,4�!4���� 4) �4$�ก�����ก*6ก�����=ก��%� 5) ��� .��-����%�3���1� ����

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27

6) ��'��!%���7%3���ก*6ก�����= 1.2 ก 7) ��'��!%���7%3���ก*6ก���!'�)�ก�� 3.3 09��ก������1�!��ก7!�0:�&��*�1� 1) ��ก ����7%3�$���4$��8�7�ก��������ก�,4� 2) ��ก �������$���� ��� ��!��!����!-* ��!7�ก�������� 3.4 1������1���(��.+ 1) ��ก �����'3$*:�%�-�!$ก&����,ก������"# 2) ��ก �����'3$*:�ก��"�$6.�!ก���'4 &3�%�8��'4��ก&����,ก������"#

4. ��� ��ก�����2����&�����2� ก!;�(�%��� �

1) 2 3�� �! G�7�3I &C1�2( 2) 2 3�� �! G �I &C1�2( 3) ���2 3D'ก"3�� 80% &C1�2( ก��*��&2�ก��&�����

"3��-4�� ก)N#&�1�"=����1� 3 ���ก��

5. ก�+ก���ก�����ก��*��0���ก��������

5.1 (�%�� � ��'�!�$�ก��%��*�*"&����� �!����"#��7%3�$�* 7�ก��%��4��*�*"� �$*:��� ����1 (�%� � 1 ���"�$ B���"�$.4$�7�6�#��= n 7%3��"�$(�ก�����.'�., �4�ก��

(�%� � 2 %��4��*�*"� ����.��(��.*�:*;&��6�#��=��ก�����.'�., &��"�$ B� %�� 3$�.��(��.*�:*;&��6�#��=��ก�����.'�., &��"�$.4$� 6�2 3 �4��*�*" =

�4� 1) an = 14n

2n3n2

2

−+

6�2 3$4� 6�#&��"�$ B���=��ก�����.'�., �8� 3n2 6�#&��"�$.4$���=��ก�����.'�., �8� 4n2

����1� lim an = lim 14n

2n3n2

2

−+ = 4

3

.��(��.*�:*;&��6�#��=��ก�����.'�., &��"�$ B� .��(��.*�:*;&��6�#��=��ก�����.'�., &��"�$.4$�

n → ∞ n → ∞

Page 28: ก 1 ˘ˇˆ˙ ก˛ - WordPress.com...ก 1 ˘ˇˆ˙ ก˛ ˚˜ ˛ !ก ก ก 1. ˛ # ก ˘. 6.1 2. ˜ % ˜ %&’ ก ( 6.1 .6/1-6 3. *˙ ก & ˙+ 3.1 ˜ ! "# - $ % & ˜ ! - ’(

28

2) an = 3n2n52n

2 −+

�8=��6�ก 3n2n52n

2 −+ =

3n2n52n0n

22

−++

6�2 3$4� 6�#&��"�$ B���=��ก�����.'�., �8� 0n2 6�#&��"�$.4$���=��ก�����.'�., �8� 2n2

����1� lim 3n2n52n

2 −+ = lim

3n2n52n0n

22

−++ = 2

0 = 0

3) an = 3n4nn 2

+−

�8=��6�ก 3n4nn 2

+− =

3n0n4nn

2

2

++

6�2 3$4� 6�#&��"�$ B���=��ก�����.'�., �8� n2 6�#&��"�$.4$���=��ก�����.'�., �8� 0n2

����1� lim 3n4nn 2

+− = lim

3n0n4nn

2

2

++

− = 01 %��4�2�42 3

5.2 (�%�*��

ก�+ก���ก�����ก��*�� <=กก��1�����

1. ��'��� .��"�$��4��ก��%��*�*"&����� �!� �$*:��� *=� "*�ก�)���=��� �! an ��'47��'(

ก�)N#��=.��

1) an = 1n42n3

−++

�8=��6�ก 3 2n + = 31

2)(n +

1n42n3

−++ = 1n

42)(n0n 31

−+++

(ก�����.'�., &��"�$ B���"�$.4$��8� ก����� 1)

����1� lim 1n42n3

−++ = lim 1n

42)(n0n 31

−+++ = 1

0 = 0

��ก��ก���* (���,ก"#7�3

n → ∞ n → ∞

n → ∞ n → ∞

n → ∞ n → ∞

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29

ก�+ก���ก�����ก��*�� <=กก��1�����

2) an = 3n25n 2

+−

�8=��6�ก 25n 2 − = 212 2)(5n −

ก�����.'�., &��"�$ B��8�ก�����&�� 212 )(n %�8�ก����� 1 ��.��(��.*�:*;&��

6�#ก�����.'�., &��"�$ B��8� 21

(5) %�8� 5

����1� lim 3n25n 2

+− = 1

5 = 5

2. ��'�:*!�� �8=����,ก�� � � &���!��*���"4�2(��1!�ก�� �� � ����� 1. �8=� a1, a2, a3, S, an (A���� �!6��ก� ���ก-�!$ก&���,ก6�#&��

��� �!7��'( a1+ a2+ a3+ S + an $4� ��,ก��6��ก�

2. �8=� a1, a2, a3, S, an, S (A���� �!����"# ���ก-�!$ก&���,ก6�#

&����� �!7��'( a1+ a2+ a3+ S + an+ S $4� ��,ก������"#

�4� 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 (A���,ก��6��ก� ��=2 36�ก��� �!6��ก� 2, 4, 6, 8, 10, 12, 14, 16 2 + 4 + 6 + ... + 2n + S (A���,ก������"#��=2 36�ก��� �!����"# 2, 4, 6, ..., 2n, S

3. ��'�:*!��.'"���=7�3%�-�!$ก n 6�#�ก&����,ก�� ����1 1) -�!$ก n 6�#�ก&����,ก�� �&�)*"

Sn = 2n [2a1+(n−1)d]

%�8� Sn = 2n [a1+ an]

2) -�!$ก n 6�#�ก&����,ก�� �&��)*"

Sn = 1r1)(ra n

1−

− , r ≠ 1 %���.��%��! |r| > 1

%�8� Sn = r1)r(1a n

1−− , r ≠ 1 %���.��%��! |r| < 1

(��=��&��.'"�&3��"3�7%3��ก ����BCก�� 3$�"� ��7�%���.8� ����%�$&3���= 1.2.1)

��ก��ก���* (��$��

n → ∞

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30

ก�+ก���ก�����ก��*�� <=กก��1�����

6�ก��1� 7%3��ก �����4��!��*���7�%���.8� ����"4�2(��1 � ����� ก��%� a1+ a2+ a3+ S + an+ S (A���,ก������"#

7%3 S1 = a1

S2 = a1+ a2

M Sn = a1+ a2+ a3+ S + an

���ก Sn $4� 2���ก���� n 0+����ก(�����ก�� �8=� n (A�6���$� "@�!$ก

���ก��� �!����"# S1, S2, S3, ..., Sn, ... $4� � ����(��2���ก����(�����ก��

(a sequence of partial sums) 4. ��'�:*!��"�$��4����= 1 ��"�$��4����= 2 7�%���.8� ���� � �7%3��ก ����"�!���D��

"4�2(��1ก4��. �$*:���� �������� � 1 1) ��,ก�� 4+7 +10+S+(3n+1)+S (A���,ก����* 7

(��,ก�� �&�)*" �8=��6�ก 4+7+10+S+(3n+1)+S (A���,ก����=2 36�ก��� �! �&�)*" 4, 7, 10, S, (3n+1), S)

2) 7%3��ก ����!�ก�4� a1 �� d

(a1 = 4

d = 7−4 = 10−7 = S = 3) 3) 7�3.'"�7 ���4$�7�ก������$)%���� �!-�!$ก�4��&����,ก��

(7�3.'"� Sn = 2n [2a1+(n−1)d] )

�������� � 2 1) ��,ก�� 21 + 4

1 + 81 + ... +

n21 + ... (A���,ก����* 7

( (A���,ก�� �&��)*" �8=��6�ก 21 + 4

1 + 81 + ... +

n21 + ... (A�

��,ก����=2 36�ก��� �! �&��)*" 21 , 4

1 , 81 , ...,

n21 , ...)

2) 7%3��ก ����!�ก�4� a1 �� r

(a1 = 21

r = 41 ÷ 2

1 = 81 ÷ 4

1 = S = 21 )

3) 7�3.'"�7 ���4$�7�ก��%���� �!-�!$ก�4��&����,ก��

(7�3.'"� Sn = r1)r(1a n

1−

�8=��6�ก |r| = 21 = 2

1 �3��ก$4� 1)

��ก��ก���* $* ����%#

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