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8.4. ALTERNATING SERIES
In this terminology, the series (8.16) converges absolutely while the alternating harmonic
series is conditionally convergent.
Activity 8.20.
(a) Consider the series
(−1)
k ln(k)
k
.
(i) Does this series converge? Explain.
(ii) Does this series converge absolutely? Explain what test you use to determine your answer.
(b) Consider the series
(−1)
k ln(k)
k 2 .
(i) Does this series converge? Explain.
(ii) Does this series converge absolutely? Hint: Use the fact that ln(k) <
√
k
for large values of k and the compare to an appropriate p-series.
⊳
Conditionally convergent series turn out to be very interesting. If the sequence {a n }
decreases to 0, but the series
a k diverges, the conditionally convergent series
(−1) k a k
is right on the borderline of being a divergent series. As a result, any conditionally
convergent series converges very slowly. Furthermore, some very strange things can
happen with conditionally convergent series, as illustrated in some of the exercises.
Summary of Tests for Convergence of Series
We have discussed several tests for convergence/divergence of series in our sections and
in exercises. We close this section of the text with a summary of all the tests we have
encountered, followed by an activity that challenges you to decide which convergence test
to apply to several different series.
Geometric
Series
The geometric series
ar k with ratio r converges
for −1 < r < 1 and diverges for |r | ≥ 1.
The sum of the convergent geometric series
∞
k=0
ar
k is a
1−r .
Divergence
Test
If the sequence a n does not converge to 0, then
the series
a k diverges.
This is the first test to apply because the conclusion
is simple.
However, if
lim n→∞ a n = 0, no conclusion can be drawn.