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.
- 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 thereexists an N, such that for every pair m, n of indices with m > n > N,,
the inequality
(4.7-1)
be satisfied.