Mathematics of Physics and Engineering

208 Power Series

Proof. If M < 1, then, by the definition of limsup, there exists age (M, 1)
and a positive integer N so that Icnl^1 /™ < q, that is, \cn\ < qn for all n> N.
Then

N-l N-l N
E M ^ E M + E <zfc = E M + rr^ < °°-
fc>0 fc=0 fe>JV fc=0 V
If M > 1, then, by the definition of limsup, there exists a q £ (1,K) so
that \cn\ > qn for infinitely many n. Then limn_>oo \cn\ ^ 0 and the series
diverges.
The following exercise completes the proof of the theorem. •

EXERCISE 4.3.4. B Construct three sequences Ylk>ock > 3 ~ 1)2,3, so

that X3fc>o cfc converges absolutely, Ylk>o ck converges conditionally, and

/Cfc>o cfc diverges, while the root test in all three cases gives M = 1.

4.3.2 Convergence of Power Series

In this section, we will see that analytic functions are exactly the functions

that can be represented by convergent power series. We start with the

general properties of power series.

A power series around (or at) point ZQ £ C is a series

^aE k(z- z 0 )k = a 0 + ai(z - zQ) + a 2 {z - z 0 )^2 H , (4.3.1)

fc>0

where ak,k > 0, are complex numbers. The following result about the

convergence of power series is usually attributed to Cauchy and Hadamard.

Theorem 4.3.3 For the power series (4-3.1), define the number R by

R = \ „. , (4.3.2)

hm sup Ja„ I^1 /"' >

with the convention 1/ + oo = 0 and 1/0 = +oo. If R = 0, then (4-3.1)

converges only for z = ZQ. If R = +oo, then (4-3.1) converges for all

zeC. If 0 < R < +oo, then (4-3.1) converges absolutely for \z — zo\ < R

and diverges for \z — ZQ\ > R.

An alternative representation for R is

JR = limsupr^-r (4.3.3)

n |a«+i|