Alternating series

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

:$sum\_\{n=0\}^infty\; (-1)^n,a\_n,$

with "a_n" ≥ 0 (or "a_n" ≤ 0) for all "n". A finite sum of this kind is an alternating sum. An alternating series converges if the terms a_n converge to 0 monotonically. The error E introduced by approximating an alternating series with its partial sum to n terms is given by |E|<|a_n+1|.

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 errorin this approximation is less than $a\_\{n+1\}$. This last observation is the basis of the Leibniz test. Indeed, if the sequence $a\_n$ tends to zero and is monotone decreasing (at least from a certain point on), it can be easily shown that the sequence of partial sums is a Cauchy sequence. Assuming

"(the sequence being monotone decreasing guarantees that $a\_\{k\}-a\_\{k+1\}>0$; note that formally one needs to take into account whether $n$ is even or odd, but this does not change the idea of the proof)"

As $a\_\{m+1\}\; ightarrow0$ when $m\; ightarrowinfty$, the sequence of partial sums is Cauchy, and so the series is convergent. Since the estimate above does not depend on $n$, it also shows that

Convergent alternating series that do not converge absolutely are examples of conditional convergent series. In particular, the Riemann series theorem applies to their rearrangements.

See also

* Nörlund-Rice integral