8.4. ALTERNATING SERIES
497
Summary
In this section, we encountered the following important ideas:
• An alternating series is a series whose terms alternate in sign. In other words, an
alternating series is a series of the form
(−1)
k a k
where a k is a positive real number for each k.
• An alternating series
∞
k=1 (−1) k a k converges if and only if its sequence {S n } of partial
sums converges, where
S n =
n
k=1
(−1)
k a k .
• The sequence of partial sums of a convergent alternating series oscillates around and
converge to the sum of the series if the sequence of nth terms converges to 0. That is
why the Alternating Series Test shows that the alternating series
∞
k=1 (−1) k a k converges
whenever the sequence {a n } of nth terms decreases to 0.
• The difference between the n − 1st partial sum S n−1 and the nth partial sum S n of a
convergent alternating series
∞
k=1 (−1) k a k is |S n − S n−1 | = a n . Since the partial sums
oscillate around the sum S of the series, it follows that
|S − S n | < a n .
So the nth partial sum of a convergent alternating series
∞
k=1 (−1) k a k approximates
the actual sum of the series to within a n .
Exercises
1. Conditionally convergent series converge very slowly. As an example, consider the
famous formula 6
π
4
= 1 −
1
3
+
1
5
−
1
7
+ · · · =
∞
k=0
(−1)
k
1
2k + 1
.
(8.17)
In theory, the partial sums of this series could be used to approximate π.
(a) Show that the series in (8.17) converges conditionally.
(b) Let S n be the nth partial sum of the series in (8.17). Calculate the error in
6 We will derive this formula in upcoming work.