Alternating Series Test

Examples 4.2.20(a):

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


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 which means for the two subsequences For each sequence we can combine pairs to see that Hence, both subsequences are monotone and bounded, and must therefore be convergent. Define their limits as Then
| 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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