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Alternating series

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

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

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

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

diverges, while the alternating version

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

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

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

converges.

The partial sum

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

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 $a_\left\{n+1\right\}$. 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 $a_\left\{k\right\}-a_\left\{k+1\right\}>0$; 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 $a_\left\{m+1\right\}rightarrow0$ when $mrightarrowinfty$, 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.