8.4. ALTERNATING SERIES
491
(c)
∞
k=2
(−1) k
ln(k)
⊳
The argument for the Alternating Series Test also provides us with a method to
determine how close the nth partial sum S n is to the actual sum of a convergent alternating
series. To see how this works, let S be the sum of a convergent alternating series, so
S =
∞
k=1
(−1)
k a k .
Recall that the sequence of partial sums oscillates around the sum S so that
|S − S n | < |S n+1 − S n | = a n+1 .
Therefore, the value of the term a n+1 provides an error estimate for how well the partial
sum S n approximates the actual sum S. We summarize this fact in the statement of the
Alternating Series Estimation Theorem.
Alternating Series Estimation Theorem. If the alternating series
∞
k=1
(−1)
k+1 a k
converges and has sum S, and
S n =
n
k=1
(−1)
k+1 a k
is the nth partial sum of the alternating series, then
∞
k=1
(−1)
k+1 a k − S n
≤ a n+1 .
Example 8.1. Let’s determine how well the 100th partial sum S 100 of
∞
k=1
(−1) k+1
k
approximates the sum of the series.
Solution.