Math 1B
§ 11.5 Alternating Series
The convergence tests we have looked at so far required that all terms in the series be positive.
An alternating series is a series whose terms are alternately positive and negative. For example,
1 1 1 1 1
1− + − + − + ...
2 3 4 5 6
−1+
1 1 1 1 1
− + − + − ...
2! 3! 4! 5! 6!
The nth term of the alternating series is of the form
an = (−1)
n−1
or
bn
an = (−1) bn
n
The Alternating Series Test: Suppose we have a series
where bn > 0 . If
i) lim bn = 0
n →∞
∑a
and either an = (−1) bn or an = (−1) bn
n
n
n−1
and
ii) {bn } is a decreasing sequence (eventually) i.e. bn +1 < bn
then the series is convergent.
Note: This will not tell us if a series will diverge.
∞
Example: Is the series
∑
n=1
Stewart – 7e
(−1)
n
n−1
convergent or divergent?
1
∞
Example: Is the series
∑
n=1
∞
Example: Is the series
∑
n=1
Stewart – 7e
(−1)
n
n2
convergent or divergent?
n2 + 5
cos( nπ )
n
convergent or divergent?
2
∞
Example: Is the series
∑
n= 0
Stewart – 7e
(−1)
n +2
n+4
n
convergent or divergent?
3