Alternating series

From Wikipedia, the free encyclopedia

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

$\sum_{n=0}^\infty (-1)^n\,a_n,$

with a_n ≥ 0. A finite sum of this kind is an alternating sum.

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

$\sum_{n=0}^\infty \frac{1}{n+1},$

diverges, while the alternating version

$\sum_{n=0}^\infty \frac{(-1)^n}{n+1}$

converges to the natural logarithm of 2.

A broader test for convergence of an alternating series is Leibniz' test: if the sequence $a n$ is monotone decreasing and tends to zero, then the series

$\sum_{n=0}^\infty (-1)^n\,a_n$

converges.

The partial sum

$s_n = \sum_{k=0}^n (-1)^k a_k$

can be used to approximate the sum of a convergent alternating series. If $a n$ is monotone decreasing and tends to zero, then the error in this approximation is less than $a n + 1$ .

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

$\sum_{n=0}^\infty (-1)^n\,a_n$

converges conditionally, then for every real number $β$ there is a reordering $σ$ of the series such that

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

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

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

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

$1-\frac12-\frac14+\left(\frac13-\frac16\right)-\frac18+\left(\frac15-\frac1{10}\right)-\frac1{12} +\left(\frac17-\frac1{14}\right)-\frac1{16}+\cdots$