Alternating series

In mathematics, an alternating series is an infinite series of the form

<math>\sum_{n=0}^\infty (-1)^n\,a_n,<math>

with a_n ≥ 0. A sufficient condition for the series to converge is that it converges absolutely. But this is often too strong a condition to ask: it is not necessary. For example, the harmonic series

<math>\sum_{n=0}^\infty \frac{1}{n+1},<math>

diverges, while the alternating version

<math>\sum_{n=0}^\infty \frac{(-1)^n}{n+1}<math>

converges to the natural logarithm of 2.

A broader test for convergence of an alternating series is the Cauchy criterion: if the sequence <math>a_n<math> is monotone decreasing and tends to zero, then the series

<math>\sum_{n=0}^\infty (-1)^n\,a_n<math>

converges.

A conditionally convergent series is an infinite series that converges, but does not converge absolutely. The following non-intuitive result is true: if the real series

<math>\sum_{n=0}^\infty (-1)^n\,a_n<math>

converges conditionally, then for every real number <math>\beta<math> there is a reordering <math>\sigma<math> of the series such that

<math>\sum_{n=0}^\infty (-1)^{\sigma(n)}\,a_{\sigma(n)}=\beta.<math>

As an example of this, consider the series above for the natural logarithm of 2:

<math>\ln 2=\sum_{n=0}^\infty \frac{(-1)^n}{n+1}=1-\frac12+\frac13-\frac14+\frac15-\cdots.

<math>

One possible reordering for this series is as follows (the only purpose of the brackets in the first line is to help clarity):

<math>1-\frac12-\frac14+\left(\frac13-\frac16\right)-\frac18+\left(\frac15-\frac1{10}\right)-\frac1{12}

+\left(\frac17-\frac1{14}\right)-\frac1{16}+\cdots<math>

<math>=\frac12-\frac14+\frac16-\frac18+\frac1{10}-\cdots<math>

<math>=\frac12\left(1-\frac12+\frac13-\frac14+\frac15-\cdots\right)<math>

<math>=\frac12\,\ln2.<math>

A proof of this assertion runs along the lines: the greedy algorithm for σ is correct.