Sequences and Series 193
and C = AB. In particular, if the factors are convergent, their prod-
uct is summable ( C, 1), and C = AB. For this and other theorems on
multiplication of series, the reader is referred to Hardy [6].
4.9 Absolute Convergence Tests
Tests for convergence of series of nonnegative real terms can be applied to
test the convergence of I:; ianJ, where the an are complex. Numerous such
tests exist. In what follows we present a summary of the most useful ones.
(a) Comparison test. First kind. Let I:;cn be a convergent series of
positive terms, I:; dn a divergent series of positive terms, and I:; an a series
of complex terms.
1. If Jani :S ken for some finite constant k > 0 and n > N (N fixed), the
series I:; Jan J converges.2. If Jani 2 kdn for some k > 0 and n > N , then I:; ianJ diverges.
Second kind.1. If ian+il/Janl :S Cn+ifcn for n > N, then I:; JanJ converges.
2. If ian+il/JanJ 2 dn+J/dn for n > N, then I:; Jani diverges.
The comparison test of the second kind is an easy corollary of the first.(b) Root test (Cauchy, 1821). First form. If for every n (or, at least
for n > N) we have Janji/n :::; r < 1 (r fixed), then I:; Jani converges. If
JanJlfn 2 1, then I:: an diverges.
Second form. If limn_,. 00 JanJ^1 fn = L, then I:; Jani converges when L <
1, and I:; an diverges if L > 1. If L = 1, the test fails. However, if
JanJlfn 2 1 for n > N, then I:; an diverges by the first form of the test. If
limn_,. 00 JanJifn does not exist (finite or infinite), then the following general
form of the root test can be applied.
Third form. If limn_,. 00 sup JanJi/n = Li, then I:; Jani converges when
Li < 1, and I:; an diverges if Li > 1. For Li = 1 the test fails. However,
if limn_,. 00 inf Jan ii/n > 1, the series I:; an diverges.
(c) Kummer's Test. First form. Let {Dn}~ be a sequence of positive
numbers, suppose that Jan J > 0 for every n, and let
.L'\n "£,. = D n -1--J Jani -D n+i
an+iThen I:; JanJ is convergent if Kn 2 k > 0 for n > N (k and N fixed), and
it is divergent if I:;(l/ Dn) diverges and Kn :S 0.
Second form. Suppose that limn--+oo Kn = L. Then I:; Jani converges if