492
8.4. ALTERNATING SERIES
If we let S be the sum of the series
∞
k=1
(−1) k+1
k
, then we know that
|S 100 − S| < a 101 .
Now
a 101 =
1
101
≈ 0.0099,
so the 100th partial sum is within 0.0099 of the sum of the series. We have discussed the
fact (and will later verify) that
S =
∞
k=1
(−1) k+1
k
= ln(2),
and so S ≈ 0.693147 while
S 100 =
100
k=1
(−1) k+1
k
≈ 0.6881721793.
We see that the actual difference between S and S 100 is approximately 0.0049750013,
which is indeed less than 0.0099.
Activity 8.18.
Determine the number of terms it takes to approximate the sum of the convergent
alternating series
∞
k=1
(−1) k+1
k 4
to within 0.0001.
⊳
Absolute and Conditional Convergence
A series such as
1 −
1
4
−
1
9
+
1
16
+
1
25
+
1
36
−
1
49
−
1
64
−
1
81
−
1
100
+ · · ·
(8.16)
whose terms are neither all nonnegative nor alternating is different from any series that we
have considered to date. The behavior of these series can be rather complicated, but there
is an important connection between these arbitrary series that have some negative terms
and series with all nonnegative terms that we illustrate with the next activity.
Activity 8.19.