11.5 Alternating Series
An alternating series is a series whose terms are alternately positive and negative. They will
often have the term (1)
n
as part of the definition of the series. The test for determining
convergence or divergence as follows:
The Alternating Series Test
If the alternating series
()
1
1234
1
n
n
bbbbb
b
−
−
=−+−+
>
∑
L
0
n
satisfies
for all
n
and
then the series is convergent.
1
nn
ab
b
+
≤
()l
im
0
n
n
bb
→∞
=
Example: Determine convergence or divergence for each series.
2
1
2
11
(1
)
l
n
)
)
1
3
n
n
n
c
n
n
∞∞
∞
−
==
=
−
−−
+
+
∑∑ ∑
1
n
n
b
Look at Example 1 on page 728. This series is the
alternating harmonic series
and it
is
convergent.
Estimating Sums
We have a theorem to use when estimating the sum of a convergent alternating series.
Theorem:
If
is the sum of an alternating series that satisfies
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 Fall '08
 Estrada
 Calculus, Mathematical Series, maximum possible error

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