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

xxexx

xe xx

PlotExpx, 1x, 1 x x^22,1 x x^22x^3123,x, 3, 3,Background RGBColor1, 1, 0

-3 -2 -1 1 2 3

-2

2

4

6

8

Out[8]= Graphics

In[17]:= PlotExpx, 1x, 1 x x^22,x, .0, 1,Background RGBColor1, 1, 0

0.2 0.4 0.6 0.8 1

1.25

1.5

1.75

2

2.25

2.5

2.75

Out[17]= Graphics

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.)