4.3. Special Series

Definition: Alternating Harmonic Series

The series is called the Alternating Harmonic series. It converges but not absolutely, i.e. it converges conditionally.


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There are many proofs of this fact. For example. the series of absolute values is a p-series with p = 1, and diverges by the p-series test. The original series converges, because it is an alternating series, and the alternating series test applies easily. However, here is a more elementary proof of the convergence of the alternating harmonic series.

We already know that the series of absolute values does not converge by a previous example. Hence, the series does not converge absolutely. As for regular convergence, consider the following two partial sums:

We have that
S 2n+2 - S 2n = 1 / (2n+1) - 1 / (2n+2) > 0
S 2n+3 - S 2n+1 = - 1 / (2n+2) + 1/ (2n+3) < 0
which means for the two subsequences
{ S 2n } is monotone increasing and { S 2n+1 } is monotone decreasing
For each sequence we can combine pairs to see that
S 2n 1 and S 2n+1 0
for all n. Hence, both subsequences are monotone and bounded and must therefore be convergent. Define their limits as
lim S 2n = L and lim S 2n+1 = M
| M - L | = | lim (S 2n+1 - S 2n) | = 1 / (2n+1)
which converges to zero. Therefore, M = L, i.e. both subsequences converge to the same limit. But this common limit is the same as the limit of the full sequence, because: given any > 0 we have Now set K = max(N, M). Then, for the above > 0 we have
| L - S n | <
for n > K because n is either even or odd. Hence, the alternating harmonic series converges conditionally.

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