Ch. 15 Power Series, Taylor Series

Home , Absolute convergence, Convergence tests, Convergent series, Divergent series, Uniform convergence

서울대학교 조선해양공학과 서유택 2017.12

※ 본 강의 자료는 이규열, 장범선, 노명일 교수님께서 만드신 자료를 바탕으로 일부 편집한 것입니다.

Seoul National 1 Univ. 15.1 Sequences (수열), Series (급수), Convergence Tests (수렴판정)

 Sequences: Obtained by assigning to each positive integer n a number zn

z . Term: zn zzzzz1212, , or , ,  or briefly   n N . Real sequence (실수열): Sequence whose terms are real

 Convergence . Convergent sequence (수렴수열): Sequence that has a limit c

lim zczc nn or simply n

. For every ε > 0, we can find N such that Convergent complex sequence

|| zcnNn  for all

→ all terms zn with n > N lie in the open disk of radius ε and center c.

. Divergent sequence (발산수열): Sequence that does not converge.

Seoul National 2 Univ. 15.1 Sequences, Series, Convergence Tests

 Convergence . Convergent sequence: Sequence that has a limit c

lim zczc nn or simply n

 Ex. 1 Convergent and Divergent Sequences iin 11 Sequence   i , , , , is convergent with limit 0. n 234

Sequence  iii,n   1,  , 1,  is divergent.

n Sequence {zn} with zn = (1 + i ) is divergent.

Seoul National 3 Univ. 15.1 Sequences, Series, Convergence Tests

 Theorem 1 Sequences of the Real and the Imaginary Parts

. A sequence z1, z2, z3, … of complex numbers zn = xn + iyn converges to c = a + ib

. if and only if the sequence of the real parts x1, x2, … converges to a

. and the sequence of the imaginary parts y1, y2, … converges to b.

 Ex. 2 Sequences of the Real and the Imaginary Parts

14 Sequence {z } with zxiyinnn 122 converges to c = 1+2i. n nn 1 4 xn 1 has the limit 1 = Re c and y n  2 has the limit 2 = Im c. n2 n

Seoul National 4 Univ. 15.1 Sequences, Series, Convergence Tests

  Series (급수):  zzzm 12 m1

. Nth partial sum: szzznn12

. Term of the series: zz12, ,

. Convergent series (수렴급수): Series whose sequence of partial sums converges 

lim ssszzz nm Then we write = 12 n  m1 . Sum or Value: s

. Divergent series (발산급수): Series that is not convergent

. Remainder: Rzzznnnn123  Theorem 2 Real and the Imaginary Parts  z A series  m with zm = xm + iym converges and has the sum s = u + iv m1

if and only if x1 + x2 + … converges and has the sum u and y1 + y2 + … converges and has the sum v.

Seoul National 5 Univ. 15.1 Sequences, Series, Convergence Tests

Tests for Convergence and Divergence of Series  Theorem 3 Divergence

If a series z + z +… converges, then lim z m  0 . 1 2 m Hence if this does not hold, the series diverges.

Proof) If a series z1 + z2 +… converges, with the sum s,

zsszssssmmmmmm11lim0 m

. zm → 0 is necessary for convergence of series but not sufficient. 111  Ex) The harmonic series 1  , which satisfies this condition but 234 diverges.

. The practical difficulty in proving convergence is that, in most cases, the sum of a series is unknown.

. Cauchy overcame this by showing that a series converges if and only if its partial sums eventually get close to each other.

Seoul National 6 Univ. 15.1 Sequences, Series, Convergence Tests

111  Ex) The harmonic series 1  diverges 234 Proof) 111111111 S 1  2345678916 1111111 1  2448888 111 1 diverge! 222

111  Ex) The harmonic series 1  converges 234

1 1 1 1 1 1 1 1 S 1          2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 11  2 2 4 4 6 6 8 8 converge!

Seoul National 7 Univ. 15.1 Sequences, Series, Convergence Tests

 Theorem 4 Cauchy’s Convergence Principle for Series

A series z1 + z2 + … is convergent if and only if for every given ε > 0 (no matter how small) we can find an N (which depends on ε in general) such that

zzznNpnnn12 p  for every and 1, 2,

Absolute Convergence (절대 수렴) . Absolute convergent: Series of the absolute value of the terms   zzzm 12 is convergent. m1

. Conditionally convergent (조건 수렴): z1+z2+… converges but |z1|+|z2|+… diverges.

1 1 1 Ex. 3 The series 1     converges but zz 12  diverges, then 2 3 4

the series z1 + z2 + … is called conditionally convergent.

Seoul National 8 Univ. 15.1 Sequences, Series, Convergence Tests

 Theorem 5 Comparison Test (비교 판정법)

If a series z1 + z2 + … is given and we can find a convergent series b1 + b2 +

… with nonnegative real terms such that |z1| < b1, |z2| < b2,…, then the given series converges, even absolutely.

Proof) by Cauchy’s principle,

bbbnNpnnn12 p  for every and 1, 2,

. From this and |z1| < b1, |z2| < b2, …,

||||zzbb nn11 pnn p 

. |z1| + |z2| + … converges, so that z1 + z2 + … is absolutely convergent.

Seoul National 9 Univ. 15.1 Sequences, Series, Convergence Tests

 Theorem 6 Geometric Series (기하 급수)  m 2 The geometric series  qqq  1 . m0 1 converges with the sum if q  1 and diverges if q  1 . 1 q

2 n Proof) sqqqn 1

21n qsqqqn  n1 sn qs n (1  q ) s n  1  q 11 qqnn11 qn1 s    since qn 1,     0 n 1q 1  q 1  q 1 q 1 s n 1 q

Seoul National 10 Univ. 15.1 Sequences, Series, Convergence Tests

 Theorem 7 Ratio Test (비 판정법)

If a series z1 + z2 + … with zn n  0  1, 2,  has the property that for every n greater than some N, z n1 q 1  n  N  zn (where q < 1 is fixed), this series converges absolutely. If for every n > N z , the series diverges. n1 1 nN zn z Proof) i) n1 1 diverges zzznn11  z 2  zn

2 ii) zznnNNNNN121321 qn N for zz qzz q z q

p1 zN p z N 1 q 1 z z  z  z(1  q  q2  )  z NNNNN1  2  3  1  1 1 q Absolutely convergence follows from Theorem 5 Comparison Test

Seoul National 11 Univ. 15.1 Sequences, Series, Convergence Tests

 Theorem 8 Ratio Test

zn1 If a series z + z + … with zn n  0  1, 2,  is such that lim  L , then: 1 2 n zn a. If L < 1, the series converges absolutely. b. If L > 1, the series diverges. c. If L = 1, the series may converge or diverge, so that the test fails and permits no conclusion.

Theorem 7 Ratio Test Proof) (a) kzzLbnnn1 /,let11 z 1 n1 qnN1   kbn 1 say11kqbn  z 2 n fornN ⇒ the series converges

Theorem 7 Ratio Test z1 + z2 + … converges k z / zLc  , let11 (b) n n1 n z n1 1 nN 1 z kcn 1 say11kcn  2 n ⇒ the series diverge Theorem 7 Ratio Test z1 + z2 +… diverges

Seoul National 12 Univ. 15.1 Sequences, Series, Convergence Tests

 Theorem 8 Ratio Test

2 111 zznn11n another series 1  2 ,,1,1as nL 4916 znznn(1)

1 111 n dx s 112        n 49 nxn221 converge!

Seoul National 13 Univ. 15.1 Sequences, Series, Convergence Tests

 Theorem 8 Ratio Test

 Ex. 4 Ratio Test Is the following series convergent or divergent? n  100 75i 1 2  1  100 75100ii 75  n0 n!2!

Sol) The series is convergent, since

100 75in1 z n1! 100 75i 125 n1     0  L z100 75in n11 n n n!

Seoul National 14 Univ. 15.1 Sequences, Series, Convergence Tests

 Theorem 9 Root Test (근 판정법)

If a series z1 + z2 + … is such that for every n greater than some N

n zqn nN 1   (where q < 1 is fixed), this series converges absolutely.

n If for infinitely many n zn  1, the series diverges.

n n Proof) (a) zqn 1 zqn nN 1 for all 1 z z  z (1  q  q2  )  1 2 3 1 q Absolutely convergence follows from Theorem 5 Comparison Test

(b) zn 1 z1 + z2 + … diverges.

This diverges from Theorem 3 Divergence.

Seoul National 15 Univ. 15.1 Sequences, Series, Convergence Tests

Caution! 111 harmonic series 1 234

n n znn 1/1 but diverge! because q < 1 is not fixed.  Theorem 10 Root Test n If a series z1 + z2 + … is such that lim zL n  , then: n a. The series converges absolutely if L < 1.

b. The series diverges if L > 1.

c. If L = 1, the test fails; that is, no conclusion is possible.

Ex) harmonic series

n n n lim1/ n limzn  lim 1/ n 1 the test fails. n nn 

Seoul National 16 Univ. 15.2 Power Series

 Power Series (거듭제곱급수)  n 2 . Power series in powers of zzazzaazzazz  0001020 :  n       n0

Coefficients: Complex (or real) constants a0, a1, …

Center: Complex (or real) constant z0  n 2 . A power series in powers of z: an zaa za012 z n0

 Convergence Behavior of Power Series  Ex. 1 Convergence in a Disk. Geometric Series  2 The geometric series  z n  1  z  z  n0 converges absolutely if z  1

diverges if z  1

Seoul National 17 Univ. 15.2 Power Series

 Convergence of Power Series . Power series play an important role in complex analysis. . The sums are analytic functions (Theorem 5, Sec. 15.3) ⇒ The sum should be convergent.

. Every analytic function f (z) can be represented by power series at z0 (Theorem 1, Sec. 15.4) ⇒ The sum should converge to f (z0).

. Ex) Maclaurin series of ez

 zzzn 23 ezz   1 n0 n!2!3!

. For specific z = z0, the left side and the right side have the same value. ⇒ The right sum should converge for the specific value. ⇒ This doesn’t mean convergence of ez as z →∞.

Seoul National 18 Univ. 15.2 Power Series

 Ex. 2 Convergence for Every z  zn z23 z The power series (which is the Maclaurin series of ez)  1 z    n0 n! 2! 3! is absolutely convergent for every z

By the ratio test, for any fixed z, n1 z z n1!   0 as n zn n! n 1

 Ex. 3 Convergence Only at the Center (Useless Series) The following series converges only at z = 0, but diverges for every z ≠ 0

  n! zzn  126 zz  23  n0 nz1! n1 (n  1) z   as n   ( z fixed and  0) nz! n

Seoul National 19 Univ. 15.2 Power Series

 Theorem 1 Convergence of a Power Series

a. Every power series converges at the center z0.

b. If a power series converges at a point zzz  10 , it converges

absolutely for every z closer to z0 than z1,that is, |z − z0| < |z1 − z0|.

c. If a power series diverges at a z = z2, it diverges for every z

farther away from z0 than z2.

 n 2  an  z z0  a 0  a 1 z  z 0  a 2 z  z 0   n0

Seoul National 20 Univ. 15.2 Power Series

 Radius of Convergence (수렴 반지름) of a Power

. Circle of convergence (수렴원): The smallest circle with center z0 that includes all the points at which a given power series converges.

. Radius of convergence (수렴반경): Radius of the circle of convergence.

|z - z0| = R is the circle of convergence and its radius R the radius of convergence.

Convergence everywhere within that circle, that is, for all z for which |z - z0| = R

Diverges for all z for which |z - z0| > R. . Notations R = ∞ and R = 0.

R = ∞: the series converges for all z.

R = 0: the series converges only at the center. Circle of convergence . Real power series: In which powers, coefficients, and center are real.

. Convergence interval (수렴구간): Interval |x-x0| < R of length 2R on the real line.

Seoul National 21 Univ. 15.2 Power Series

 Theorem 2 Radius of Convergence R

Suppose that the sequence aan nn  1 /, 1, 2,  , converges with limit L*. If L* = 0, then R = ∞; that is, the power series converges for all z. If L*  0 (hence L* > 0), then 1 a R lim n Cauchy - Hadamard formula n La* n1

If aa nn  1 /  , then R = 0 (convergence only at the center z0) Proof) The ratio of the terms in the ratio test is n1 azzann101() an1 = zz 0 LzzLlim* zz 00LLzz=*  0 n n ann() zza 0 an i) Let L* ≠ 0, thus L* > 0

The series converges if L* = L*|z − z0| < 1, |z − z0| < 1/L* and diverge if |z − z0| > 1/L*. ⇒ 1/L* is radius of convergence. ii) If L* = 0, then L = 0 for every z. ⇒ convergence for all z by the ratio test. a iii) If , then n  1 ⇒ diverge for any z ≠ z and all sufficiently zz0 1 0 large n. an

Seoul National 22 Univ. 15.2 Power Series

 Theorem 2 Radius of Convergence R

If aa nn  1 /  , then R = 0 (convergence only at the center z0)

Ex. 5 Radius of Convergence  2!n n Radius of convergence of the power series  2 zi3  n0 n!

2!n 2 2 Sol) n!2 2nn !n 1!    1 1 R lim  lim  lim  n2n 2 ! n 2 n  2 2n 2 !n!  2 n  2 2 n  1 4 n1!   1 1 The series converges in the open disk zi  3 of radius and center 3i. 4 4

Seoul National 23 Univ. 15.2 Power Series

 Radius of Convergence R 1 a 1 n n R lim R, L lim an n n La* n1 L

Ex. 6 Extension of Theorem 2 Find the radius of convergence of the power series.   11111 1 (  1)322nn zzzzz 235  n  n0  224816 111 Sol) The sequence of the ratios aa nn  1 /, 2(2),  644 1/ 8(2) does not converge. Thus, we can’t use Theorem 2 for this example. 1 For odd n, n a =1/n 2 n= n 2 For even n, n an =n 2 1/ 2n  1 as   1 n R  l , l : the greatest limit point of the sequence

R 1 , that is, the series converge for z  1 .

Seoul National 24 Univ. 15.2 Power Series

 Radius of Convergence R 1 a 1 n n R lim R, L lim an n n La* n1 L

Ex) Find the center and the radius of convergence  (2)!n (2)zi n  n 2 n0 4(!)n

Sol)

2!n n1 2 2 4!n n 2 2nn !441 ! 1n       R limlimlim1 nnn2n 2 ! 2nnn 2 !2 2n 22 1 n1 2   4!n     4n 1 ! 

Seoul National 25 Univ. 15.3 Functions Given by Power Series

 Terminology and Notation  n Given power series  az n has a nonzero radius of convergence R (thus R > 0) n0 → Its sum is a function of z, say f (z)

 n 2 f za  zaa zan zzR 012   n0

→ f (z) is represented by the power series.

 Uniqueness of a Power Series Representation A function f (z) cannot be represented by two different power series with the same center.

Seoul National 26 Univ. 15.3 Functions Given by Power Series

 Theorem 1 Continuity of the Sum of a Power Series If a function f (z) can be represented by a power series with radius of convergence R > 0, then f (z) is continuous at z = 0.

 Proof) n 2 f za  zaa zn a zz 0120 r Rfa converges for | | 0   n0

We must show that lim fza   0 z0 For a given   0 Q: If not continuous?

There is a δ > 0 such that z   implies fza    0 

 nn1 1 ann ra rS nn00r when/  S

for 0<|z| r We can always find a δ > 0 such that ||/zS which implies f  z   a 0  .

   nnn1 1 fza()(/)0  azn  zaz  n  zar  n  zSS    SS   n1 n  1 n  1

Seoul National 27 Univ. 15.3 Functions Given by Power Series

We can find zS   / which satisfies fza  0 

a0+ε Case I a a +ε  / S 0 0 a0−ε

a0 a0−ε −ε/S −δ δ ε/S

−δ δ Case II a +ε  / S 0

For a given   0 a0 a −ε If for z   0

then for any   z    , is always satisfied. −δ −δ δ δ

but for any z   , -ε/S ε/S is not always satisfied. We can find a new   / S which satisfies f  z   a 0  .

Seoul National 28 Univ. 15.3 Functions Given by Power Series

 Theorem 2 Identity Theorem for Power Series. Uniqueness 2 2 Let the power series a0 + a1z + a2z + … and b0 + b1z + b2z + … both be convergence for |z| < R, where R is positive, and let them both have the same sum for all these z.

→ Then the series are identical, that is, a0 = b0, a1 = b1, a2 = b2, … Hence if a function f (z) can be represented by a power series with any

center z0, this representation is unique.

Proof) We proceed by induction (귀납법). By assumption,

2 2 a0 + a1z + a2z + … = b0 + b1z + b2z + …

The sums of these two power series are continuous at z = 0 → a0 = b0. m+1 Now assume that an = bn. For n = 0, 1, …, m. Divide the result by z

2 2 am+1 + am+2z + am+3z + … = bm+1 + bm+2z + bm+3z + …

Letting z → 0, we concluded form this that am+1 = bm+1.

Seoul National 29 Univ. 15.3 Functions Given by Power Series

 Operations on Power Series . Termwise addition or subtraction

Termwise addition or subtraction of two power series with radii of convergence R1 and R2.

→ A power series with radius of convergence at least equal to the smaller of R1 and R2.

. Termwise multiplication Termwise multiplication of two power series   n and n fza  zaa z n 01 gzb  zbb z n 01 n0 n0

Cauchy product of the two series:  n 2 ab0n ababz 1 nn  1000  01ab  1002 ab abz  11  20 ab ab abz  n0

. Termwise differentiation and integration by termwise differentiation, that is,  n12 nan z a1 23 a 2 z  a 3 z  n1

Seoul National 30 Univ. 15.3 Functions Given by Power Series

 Theorem 3 Termwise Differentiation of a Power Series The derived series (미분급수) of a power series has the same radius of convergence as the original series.

1 a Proof) n nan n aann R lim limlimlimlim  La* n nnnn n1 (1)(1)nanaannn111  nn1 Ex. 1 Application of Theorem 3 aznaznn nn01 Radius of convergence of the power series n   n(1)(1) nn k  n nnnn( 1) 2345 nkC zz zzzz  3610. k k! 2 nn22 2! n!    n2 ()!n ! k k Sol) f zzzz  n 1 23 fzn  nz (1)  n0 n2 a (nn 1) R lim1 n R lim 1 n n an1 nn( 1) z2 n( n  1) n (nn 1) f z znn z R lim  1    n nn( 1) 22nn222

Seoul National 31 Univ. 15.3 Functions Given by Power Series

 Theorem 4 Termwise Integration of Power Series  a aa The power series n n123 12  za zzz0 n0 n 123 2 obtained by integrating the series a0 + a1z + a2z + … term by term has the

same radius of convergence as the original series.

Seoul National 32 Univ. 15.3 Functions Given by Power Series

 Power Series Represent Analytic Functions “거듭제곱급수는 해석함수다”

 Theorem 5 Analytic Functions. Their Derivatives . A power series with a nonzero radius of convergence R represents an analytic function at every point interior to its circle of convergence.

. The derivatives of this function are obtained by differentiating the original series term by term.

. All the series thus obtained have the same radius of convergence as the original series.

. Hence, by the first statement, each of them (도함수) represents an analytic function.

Seoul National 33 Univ. 15.3 Functions Given by Power Series

 Ex) Find the radius of convergence in two ways: (a) directly by the Cauchy– Hadamard formula in Sec. 15.2, and (b) from a series of simpler terms by using Theorem 3 or Theorem 4. 1 a R limn Cauchy - Hadamard formula 1 n  La* n1 nk nk z n n(1)(1)! nnkn n0 k  k knkk!()!! 

11 Sol) (a)  nknkn kkn   zzz   kk nn00  an

k n k   n 1  k  z /     a kk(n 1  k )!/!( k n  1)! n !( n  1  k )! n  k  1 n        1 an1 k n1  k   n  k  ( n k )!/!! k n ( n  k )!( n  1)! n  1 z /     kk    a R limn 1 n an1

Seoul National 34 Univ. 15.3 Functions Given by Power Series

 Ex) Find the radius of convergence in two ways: (a) directly by the Cauchy– Hadamard formula in Sec. 15.2, and (b) from a series of simpler terms by using Theorem 3 or Theorem 4.

1  nk nk n n(1)(1)! nnkn z  n0 k k knkk!()!! 

1  Sol) (b) nk n k(nk )! ! k  n 1 k  n z:! z k z n0 k (n k )! ( n  1)( n  2) ( n  k ) Integrate k times term by term kz! k

∴ Radius of convergence = 1

Seoul National 35 Univ. 15.4 Taylor and Maclaurin Series

Taylor series Taylor series of a complex function f (z):

 n 11n fz * f za  z zafzdz  where *      nn00ni!2  n1 n1 C zz* 0 

Integrate counterclockwise around a simple closed path C that contains z0 in its interior. f (z) is analytic in a domain containing C and every point inside C.

 Maclaurin series: Taylor series with center z0 = 0  Taylor’s formula 2 n zz  z zz z   n fz   fzfzfzf   0  z Rz 00        0000 1!2!! n n Remainder:  z zn1 f z * R z  0 dz * n 2i  n1 C  z** z0   z z

Seoul National 36 Univ. [Reference] 14.4 Derivatives of Analytic Functions

 Theorem 1 Derivatives of an Analytic Function If f (z) is analytic in a domain D, then it has derivatives of all orders in D, which are then also analytic functions in D. The values of these

derivatives at a point z0 in D are given by the formulas 1 fz  fzdz   0 2i  2 C  zz 0  2! fz  fzdz   0   3 2i C  zz  and in general 0

n n! fz  f   z  dz,  n 1, 2,  0 2i  n1 C zz 0 

here C is any simply closed path in D that enclose z0 and whose full interior belongs to D; and we integrate counterclockwise around C.

Seoul National 37 Univ. 15.4 Taylor and Maclaurin Series

 Theorem 1 Taylor’s Theorem

. Let f (z) be analytic in a domain D, and let z = z0 be any point in D.

. Then there exists precisely one Taylor series with center z0 that represents f (z).

. This representation is valid in the largest open disk with center z0 in which f (z) is analytic. The remainders Rn(z) of the power series can be represented in the form  z zn1 f z * R z  0 dz * n 2i  n1 C  z** z0   z z M a  . The coefficients satisfy the inequality n r n

. where M is the maximum of |f (z)| on a circle | z − z0 | = r in D whose interior is also in D.

Seoul National 38 Univ. 15.4 Taylor and Maclaurin Series

Cauchy’s Integral Formula 1 fz *  Proof) fzdz   * fz 2*izz  1   C   fzdz   0 2izz  1 1 1 C 0  zz  z**() z z  z  z  z zz 0 1 00(zz * ) 1 0 0 zz* 0 zz* 0 = q 11 qqnn11 1...qqn 111qqq 11qn1  1 qq  ... n   11qq n f z  an  z z0  n1

zz 11n fz * q  0 a f   z dz * n ni!20   n1 zz* C  zz* 0  21nn 111 z zz zz zz z  1 0000  zzzzzz******* zzzzzzzz 000000

11f z**** f z z z f z () z z n f z  f z  dz****()  dz 00 dz   dz  R z 2i z * z 2  i  z * z 2  i 2 2  i  n n CCCC   0   z** z00  z z 

Seoul National 39 Univ. 15.4 Taylor and Maclaurin Series

 Proof-continued 1 f zf*** zf z z zz z   ()n   f zdzdzdzR   z ***( ) 00 2*22izzii  2 n n CCC 0   zzzz**00    n f z  an  z z0  will converge and represent f (z) if and only if n1

limRzn    0 n

f (z) is analytic inside and on C → f (z*)/(z*- z) is analytic inside and on C. fz *  M  f z dzML  ML - inequality C  zz* 0  nn11 z zfz * z z 1 zz n1 R z 00 dz*2  M r  M0 r n 22 n1 rrn1 C  zz* 0 

z z00  r  z  z/1 r 

lim Rn  z 0 n

Seoul National 40 Univ. 15.4 Taylor and Maclaurin Series

 Accuracy of Approximation. We can achieve any preassigned accuracy in approximating f(z) by a partial sum by choosing n large enough.

 Singularity, Radius of Convergence. . Singular point: Point at which the function is not analytic . On the circle of convergence there is at least one singular point (z = c) . The radius of convergence R is usually equal to the distance from the center (z0) to the nearest singular point. singular point zc

 Theorem 2 Relation to the Last Section A power series with a nonzero radius of convergence is the Taylor series of its sum. Power series Taylor series   n 2 n 1 n  an  z z0  a 0  a 1 z  z 0  a 2 z  z 0   f z  ann z  z00 where a  f z  n0 n1 n!

Seoul National 41 Univ. Derivatives of Analytic Functions nfz!() 15.4 Taylor and Maclaurin Series fzdz()n () 0 C n1 2()izz  0

 Maclaurin series