## 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.

### Proof:

*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:

and

We have that

andS_{2n+2}- S_{2n}= 1 / (2n+1) - 1 / (2n+2) > 0

which means for the two subsequencesS_{2n+3}- S_{2n+1}= - 1 / (2n+2) + 1/ (2n+3) < 0

For each sequence we can combine pairs to see that{ Sis monotone increasing and_{2n}}{ Sis monotone decreasing_{2n+1}}

for allSand_{2n}1S_{2n+1}0

*n*. Hence, both subsequences are monotone and bounded and must therefore be convergent. Define their limits as

Thenlim Sand lim S_{2n}= L_{2n+1}= M

which converges to zero. Therefore,| M - L | = | lim (S_{2n+1}- S_{2n}) | = 1 / (2n+1)

*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

- there exists an integer
*N*such that*| L - S*if_{2n}| <*n > N* - there exists an integer
*M*such that*| L - S*if_{2n+1}| <*n > M*

*K = max(N, M)*. Then, for the above

*> 0*we have

for| L - S_{n}| <

*n > K*because

*n*is either even or odd. Hence, the alternating harmonic series converges conditionally.