1 chapter 1 infinite series, power series ( 무한급수, 멱급수 ) mathematical methods in the...
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1
Chapter 1 Infinite series, Power series ( 무한급수 , 멱급수 )
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Lecture 1 Infinite series, convergence
2
1. Geometric series ( 기하급수 )
13
2 0,,
3
2
3
21,,
27
8,
9
4,
3
2,1
12 ,,222,,16,8,4,2
,,,,,,
11
1
132
r
r
arararara
nn
nn
n
- Geometric sequence ( 기하수열 )
- Series ( 급수 ): an indicated sum of a given sequence.
cf. Infinite series, Geometric series ( 무한급수 , 기하급수 )
1
1
3
2
27
8
9
4
3
21
2216842n
n
3
- Summation of the geometric series, S_n
r
raSraSr
ararSS
arararararrS
ararararaS
ara
n
nn
n
nnn
nnn
nn
nn
1
11)1(
:sequence Geometric
132
132
1
g"oscillatinor divergent " ,1For
"convergent" 0lim 1
lim ,1For
r
rr
aSSr n
nn
n
- Summation of the infinite geometric series
4
Example) Traveling distance of bouncing ball
ov
oevv 1
ovev 22
ovev 33
ovev 44
on
n vev
2
20
220
2
20
20
10
12
21
222
21
1
1
21
1
2
)(2
22
222
1
1 ,
eg
ve
g
v
eg
ev
g
vhhh
heg
ve
g
vhmvmgh
eevv
nntotal
nnn
nnn
nn
5
3. Application of series ( 급수의 응용 )
- It is possible for the sum of an infinite series to be nearly the same as the sum of a fairly small number of terms at the beginning of the series. Many applied problems can not be solved exactly, but we may be able to find an answer in terms of an infinite series, and then use only as many terms as necessary to obtain the needed accuracy.
- There is more to the subject of infinite series than making approximations. We will see how we can use power series (that is, series whose terms are powers of x) to give meaning to functions of complex numbers.
- Other infinite series: Fourier series (sines and cosines), Legendre series, Bessel series, and so on.
1for 1!3!2
1 ex.32
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4
6
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7
4. Convergent and divergent series ( 수렴급수 , 발산급수 )
number). (finite lim SSnn
a. If the partial sum S_n of an infinite series tend to a limit S, the series is called convergent. Otherwise, it is called divergent.
b. The limiting value S is called the sum of the series.
c. The difference R_n=S-S_n is called the remainder (or the remainder after n terms).
.convergent is series theif ,0limlim
SSSSR nn
nn
cf. Convergent and Divergent series (Summation)
nn ararararara 132
to get summation when converged to check if it converges or diverges
8
5. Testing series for convergence; the preliminary test ( 수렴에 대한 검사 ; 예비검사 )
- Preliminary test (divergence condition)
a. If the terms of an infinite series do not tend to zero, the series diverges.
b. If they do, we must test further.
5
4
4
3
3
2
2
1 ex.
diverges. ,0lim nn
naa
9
6. Convergence tests for series of positive terms; absolute convergence ( 양의 항으로 이루어진 급수에 대한 수렴검사 ; 절대수렴 )
If some of the terms of a series are negative, we may still want to consider the
related series which we get by making all the terms positive; that is, we may
consider the series whose terms are the absolute values of the terms of our
original series.
If the new series converges, we call the original series absolutely convergent.
It can be proved that if a series converges absolutely, then, it converges.
10
A. The comparison test ( 비교검사 )
1) smaller than the convergent series
. if ,convergent absolutely be should
convergent :
321
321
nn maaaa
mmm
2) larger than the divergent series
. if diverges, should
divergent :
321
321
nn daaaa
ddd
11
B. The integral test ( 적분검사 )
diverges. infinite, is If .
converges. finite, is If .
0 1
nn
nn
nn
adnab
adnaa
aa
Example. .ln1
.4
1
3
1
2
11
ndnn
12
C. The ratio test ( 비율검사 ): in cases that we cannot evaluate the integral.
anything. us not tell does test ratio the1, If
diverges. 1 with series aA
converges. 1 with series aA
.lim,1
nn
n
nn a
a
Example
01
1limlim,
1
1
!
1
!1
1
!
1
!3
1
!2
11
nnnn
n
nn
nn
So, the above series should be convergent.
13
D. A special comparison test (may skip this.) ( 특별 비교검사 )
1n
1
1n
1
diverges.
)(or 0an greater thlimit a/ and
,0 terms,positive of seriesdivergent :
converges.
limit finite a/ and
,0 terms,positive of series convergent :
n
nn
nn n
n
nn
nn n
a
ba
ad
a
ba
ab
14
7. Alternating series ( 교대급수 )
- Alternating series: a series whose terms are alternatively plus and minus.- Test for alternating series: An alternating series converges if the absolute value of the terms decreases steadily to zero.
0lim,1 nnnn aaa
01
lim,1
1
1
converges, :1
4
1
3
1
2
11
1
nnn
n
n
n
Example.
15
8. Conditionally convergent series ( 제한적인 수렴급수 )
- A series converges, but does not converge absolutely. In this case, it is a conditionally convergent series.
9. Useful facts about series ( 급수에 대한 유용한 사실 )
1. The convergence or divergence of a series is not affected by multiplying
every term of the series by the same nonzero constant. Neither is it affected by
changing a finite number of terms (for example, omitting the first few terms.)
2. Two convergent series may be added (or substracted) term by term. The
resulting series is convergent, and its sum is obtained by adding the sums of
the two given series.
3. The terms of an absolutely convergent series may be rearranged in any order
without affecting either the convergence or the sum.
Chapter 1 Infinite series, Power series
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Lecture 2 Power series (Taylor expansion)
10. Power series ( 멱급수 )
nn
n
nn
n
nn
n
nn
xaxaxS
axaaxaxaxaaaxa
xaxaxaaxa
convergentfor
or
0
33
2210
0
33
2210
0
cf. interval of convergence ( 수렴 구간 )
11. Theorems about power series ( 멱급수에 대한 정리 )
Theorem 1. A power series may be differentiated or integrated term by term; the
resulting series converges to the derivative or integral of the function represented
by the original series within the same interval of convergence as the original
series.
Theorem 2. Two power series may be added, subtracted, or multiplied; the
resolution series converges at least in the common interval of convergence.
Theorem 3. One series may be substituted in another provided that the values of
the substituted series are in the interval of convergence of the other series.
Theorem 4. The power series of a function is unique, that is, there is just one
power series which converges to a given function.
12. Expanding function in Power series ( 함수의 멱급수 전개 )
(using the differentiation…)
021
0,0at ,43322sin
) ( derivative Second iii)
1,0at ,432cos
) ( derivativeFirst ii)
00sin,0At i)
sin
this.likeset First
22
432
13
42
321
0
44
33
2210
axxaxaax
axxaxaxaax
axx
xaxaxaxaxaax nn
미분계수두번째
미분계수첫번째
origin) about the seriesTalyor :series (Maclaurin
!6!4!2
1cos cf.
!7!5!3sin
series.power theof tscoefficien thedeterminecan we way,In this
!3
1
321
1,0at ,43232cos
) ( derivative Third iv)
642
753
343
xxxx
xxxxx
axxaax
미분계수세번째
- General Talyor series for f(x)
axnannxf
axnnnaxf
axnnaxaaxf
axnaaxaaxaaxf
axaaxaaxaaxaaxaaxf
nn
n
n
nn
nn
of powers containing terms12321
1232
1322
32
33
3
232
12321
44
33
2210
0!
0!3
0!2
00,0For
!
1
!3
1
!2
1
!,!332,!22,,
,For
332
332
333
2210
nn
nn
nn
fn
xf
xf
xfxfxfx
afaxn
afaxafaxafax
afxf
anafaaafaaafaafaaf
ax
13. Techniques for obtaining power series expansions ( 멱급수 전개를 얻는 방법 )
There are often simpler ways for finding the power series of a function than
the successive differentiation process in Section 12. Theorem 4 in Section 11
tells us that for a given function there is just one power series. Therefore we can
obtain it by any correct method and be sure that it is the same Maclaurin series
we would get by using the method of Section 12. We shall illustrate a variety of
methods for obtaining power series.
1
!3
21
!2
111
11 432
11ln
!4!3!21
!
!6!4!21
)!2(
1cos
!7!5!3)!12(
1sin
32
0
432
0
1
432
0
642
0
2
753
0
12
xxppp
xpp
pxxn
px
xxxx
xn
xx
xxxx
n
xe
xxx
n
xx
xxxx
n
xx
n
np
n
nn
n
nx
n
nn
n
nn
First, please memorize these basic series for your timesaving.
A. Multiplication ( 곱하기 )
Ex.1
!3!3
!5!31sin1
432
53
xxxx
xxxxxx
Ex. 2
631
!4!21
!4!3!21cos
43
42432
xxx
xxxxxxxex
xiixxx
ixix
eeexe
xexixe
1ReRecos
cosRe ,sincos cf.
B. Division ( 나누기 )
Ex.1
4321
111
432
11ln
1
32
0
11
0
1432
xxx
n
x
n
x
x
xxxx
xx
x n
nn
n
nn
Ex.2
32
3
32
2
2
1
1
111
1 xxx
x
xx
x
xx
x
x
xx
53
5342
15
2
!3
!5!3!4!2
1cos
sintan
xx
x
xxx
xx
x
xx
Ex. 3
C. Binomial Series ( 이항 급수 )
ex. .1
32
321
1
!3
321
!2
2111
1
1
xxx
xxxxx
!
121
!2
1
2,
1,1
0
n
npppp
n
p
pppp
pp
D. Substitution of a polynomial or a Series for the variable in another series ( 다항식이나 급수를 다른 급수의 변수로 바꾸어 넣기 )
ex. 1
!4!3!21
!4!3!21
!4!3!21
8642
4232222
4322
xxxx
xxxx
XXXXee Xx
ex.2
432
3
532
53
53
432tan
8
3
221
!3
15
2
3
!2
15
2
3
15
2
31
!4!3!21
xxx
x
xx
xxx
x
xx
x
XXXXee Xx
E. Combination of methods ( 방법들의 결합 )
ex. How to expand arctan x in a Taylor series
x txt
t
dt0 02
.arctanarctan1
.753
arctan
7531
1
11
753
753
0
642
0 2
64212
xxxxx
xxxxdtttt
t
dt
tttt
xx
F. Using the basic Maclaurin series (x=0) ( 기본적인 Maclaurin 급수의 사용 )
ex. 1 432 14
11
3
11
2
1111lnln xxxxxx
ex. 2
.2
3
!5
1
2
3
!3
1
2
3
2
3sin
2
3
2
3coscos
53
xxx
xxx
G. Using a computer ( 컴퓨터 사용 )
30631,
631,
31,1
30631cos
543
5
43
4
3
31
543
xxxxS
xxxS
xxSxS
xxxxxex
-5 -4 -3 -2 -1 0 1 2 3 4 5
-40
-20
0
20
40 Original S1 S3 S4 S5
33
Chapter 1 Infinite series, Power series
Mathematical methods in the physical sciences 3rd edition Mary L. Boas
Lecture 3 Application of Power series
34
15. Some uses of series ( 급수의 활용 )
1) Numerical computation ( 수치 계산 ) : With computers and calculators so available, you may wonder why we would ever want to use series for numerical computation. Here is an example to warn you of the pitfalls of blind computation.
ex.1
0015.0
tan1
1ln
x
xx
x
16
0015.0
75
753753
1006.5~45
4
15
315
17
15
2
3753
x
xx
xxx
xxxx
x
error order, x^7~10^-21
35
ex.2
!5!3
1sin
1 1062
4
4
1.0
24
4 xxx
xdx
dx
xdx
d
x
36
2) Summing series ( 급수 더하기 )
1
432
4321ln
4
1
3
1
2
11
x
xxxxx
3) Evaluation of definite integral ( 구간 적분의 계산 )
ex.
31028.0
00076.002381.0333.0
!511
1
!37
1
3
1
!5!3sin
1
0
10621
0
2
dx
xxxdxx
37
4) Evaluation of indeterminate forms ( 부정 형태의 계산 )
x
ex
x
1lim
01
!21lim
!3!211
lim0
32
0
x
x
xxx
xx
cf. L’Hopital’s rule
.0,0 if ,limlim
agagafxg
xf
xg
xfaxax
ex.
.11
lim1
lim1
lim000
x
x
x
x
x
x
e
x
e
x
e
cf: confer (= compare)e.g.: exempli gratia (= for example)i.e.: id est (= that is)
38
xg
xf
g
f
xgxgg
xfxff
xgxgxg
xfxfxf
xgxgxgg
xfxfxff
xg
xf
xx
x
xx
02
2
0
32
32
0
32
32
00
lim0
0
0!310!210
0!310!210lim
0!310!210
0!310!210lim
0!310!2100
0!310!2100limlim
proof)
39
5) Series approximation ( 급수의 어림 )
ex.1 Equation of motion of a simple pendulum
kxF
2
22
2
2
2sinsin
sin
force) (storing
Tm
ktAmtkA
tAx
xmkx
xmdt
dmmaF
kxF
40
l
mgsinmg cosmg
solve' todifficult '0sin
sin
sin
l
g
mlmg
mgF
mlxmmaF
oscillator harmonic simple : )1for sin( 0 l
g
41
15-28 Special relativity ( 특수 상대성 )
21
2
2
0
21
2
220 1.1
c
vmmcfc
vcmE
2
221
2
2
2
2
2
222121
2
2
2
111,1For
2
11
4
3
2111
cv
cv
cv
cvX
XX
cv
20
202
220
21
2
220 2
1
2
111 vmcm
cvcm
cvcmE
rest mass energy
kinetic energy
42
Homework
Chapter 1 1-1 6-8, 6-22 13-7, 13-27 (15-27, 28, 29, 31, and 33: You do not have to do these as a homework. But, it is likely that one of them will show up in the midterm exam.)
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