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716 CHAPTER 21 Series Representations of Functions

converges, then lim n→∞ a n =lim n→∞ b n =0, so lim n→∞ c n =0 also. As with real series, the general
term c n of a convergent complex series must have a limit of 0 as n →∞.
We say that ∞ c n converges absolutely if the real series ∞ |c n | converges. As with real
n=1 n=1
series, absolute convergence of a complex series implies its convergence. For suppose ∞ |c n |
n=1
converges. Since |a n |≤|c n |, then ∞ |a n | converges by comparison, so ∞ a n converges.
n=1 n=1
Similarly, |b n |≤|c n |,so ∞ b n converges, and therefore, ∞ c n converges.
n=1 n=1
Power Series and Taylor Series

A power series is a series of the form
∞
n 2
c n (z − z 0 ) = c 0 + c 1 (z − z 0 ) + c 2 (z − z 0 ) + ··· .
n=0
The complex numbers c n are the coefficients of the power series, and z 0 is its center.A
fundamental issue about any power series is determination of those values of z for which it
converges. We will show that, if a power series converges at some point z 1 different from
z 0 , then it must converge absolutely at all points closer to z 0 than z 1 .

THEOREM 21.1 Convergence of Power Series
∞
n
Suppose c n (z − z 0 ) converges at z 1 different from z 0 . Then this series converges absolutely
n=0
at all z satisfying
|z − z 0 | < |z 1 − z 0 |.
n
Proof Because ∞ c n (z 1 − z 0 ) converges,
n=0
n
lim c n (z 1 − z 0 ) = 0.
n→∞
This means that we can make the terms of the series as small in magnitude as we like by choosing
n to be large enough. In particular, for some N,
n
|c n (z 1 − z 0 ) | < 1if n ≥ N.
Then, for n ≥ N,
n

(z − z 0 )
n n
|c n (z − z 0 ) | = |c n (z 1 − z 0 ) |
(z 1 − z 0 )
n

n
(z − z 0 )
≤
(z 1 − z 0 )
n
n
z − z 0
= < 1
z 1 − z 0

because |z − z 0 | < |z 1 − z 0 |. Then the geometric series
∞ n
z − z 0

z 1 − z 0
n=0
converges. By comparison,
∞

n
|c n (z − z 0 ) |
n=N
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October 14, 2010 15:35 THM/NEIL Page-716 27410_21_ch21_p715-728