文档介绍：Section
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Infinite Series:
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 £© Absolute Convergence
At this point we have limited our study of series primarily to those series having nonnega-
tive terms, the only exceptions being some geometric series and series which are multiples
of series with nonnegative terms. In this section we shall consider the more general question
of series with negative as well as positive terms.
An important consideration when looking at the behavior of an arbitrary series
∞
a ()
X n
n=1
is the behavior of the related series
∞
|a |. ()
X n
n=1
Of course, if all the terms of () are nonnegative, then () and () are the same
series. In any case, () has all nonnegative terms, so we may use our results of the
last three sections to help determine whether or not it converges. Suppose that, by one
method or another, we have shown that () converges. Then, since
0 ≤ an + |an| ≤ 2|an| ()
for any n, we know, by parison test, that the series
∞
(a + |a |) ()
X n n
n=1
converges. Hence
∞∞∞
a = (a + |a |) −|a | ()
X n X n n X n
n=1 n=1 n=1
converges since it is the difference of two convergent series. That is, the convergence of
() implies the convergence of ().
∞∞
Proposition If |an| converges, then an converge