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Notes on the Alternating Series Test
The Alternating Series Test applies only to a series with a very special form.
Namely, the series must be of the form
n
a
∑
, where
1
0
n
n
a a
+
<
. That is, the terms of the
series alternate between positive and negative values. The test is relatively easy to apply
and one can estimate the error between the actual sum of the series and a partial sum of
the series.
The Alternating Series Test
: Let
(
29
1
n
n
b

∑
be an alternating series with
0
n
b
. The
series converges if
a)
1
n
n
b
b
+
≤
for all
n
, and
b)
lim
0
n
n
b
→∞
=
.
There are several observations one must keep in mind in applying the Alternating
Series Test.
A. This is the first test we have that applies to a series where the terms are not
eventually all positive. But one must identify the terms
n
b
so that they are
positive.
B. One must verify Condition (a) when applying this test. While the test
states the inequality holds for
all n
, it need only hold for
all n sufficiently
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 Spring '11
 BUEKMAN

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