1550251515-Classical_Complex_Analysis__Gonzalez_

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186 Chapter4


From a more technical (but less intuitive) standpoint, a series is defined
as an ordered pair of sequences, namely, (an, Sn) where Sn is defined as
in ( 4.6-2).


Examples (1) The series I::=l [ n(,;+l) 1 21 n i] converges to 1 + i. In


fact, we have


Sn= ( 1 \ + ~ i) + ( 2 \ + ;2 i) + ... + [ n(n ~ 1) + 2


1
n i]

[ 1 \ + 2 \ + ... + n(n


1

+ 1)] + ( ~ + ;2 + · · · + 2


1
n) i

= (1 -_l ) + (1 -]___) i


n+ 1 2n


SO that limn->oo Sn = 1 + i.



  1. The series I::=i [n + ( -1 riJ diverges. Here


Sn= (1 - i) + (2 + i) + · · · + [n + (-ltiJ
= (1+2 + ... + n) +^1 /2[-l + (-lrJi
=^1 / 2 n(n + 1) +^1 / 2 J-l + (-1rJi
which does not tend to a finite limit as n -> oo.

Definition 4.9 The series I:~ an is said to be absolutely convergent iff

I:~ \an\ converges.

4.7 Criteria for Convergence of Series of Complex Numbers

As in the case of numerical real series, it is often impossible to decide on
the convergence (or divergence) of a given series on the basis of Defini-
tion 4.8. In this section we discuss several criteria for convergence based
on knowledge of the sequence {an} of the terms of the series. We begin by
considering the Cauchy criteria as applied to series, which is particularly
useful in theoretical work.

Theorem 4.5 (Cauchy Condition for Series). A necessary and sufficient

condition for a series I:~ an to be convergent is that for every e > 0 there

exists an N, such that for every pair m, n of indices with m > n > N,,


the inequality
(4.7-1)
be satisfied.
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