Alternating series

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

\sum_{n=0}^\infty (-1)^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_{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 an 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 an is monotone decreasing and tends to zero, then the error in this approximation is less than an + 1. This last observation is the basis of the Leibniz test. Indeed, if the sequence an 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 m < n,

\begin{array}{rcl} \displaystyle\left|\sum_{k=0}^m(-1)^k\,a_k\,-\,\sum_{k=0}^n\,(-1)^k\,a_k\right|&=&\displaystyle\left|\sum_{k=m+1}^n\,(-1)^k\,a_k\right|=a_{m+1}-a_{m+2}+a_{m+3}-a_{m+4}+\cdots+a_n\\ \ \\&=&\displaystyle a_{m+1}-(a_{m+2}-a_{m+3}) - (a_{m+4}-a_{m+5}) -\cdots-a_n<a_{m+1} \end{array}


(the sequence being monotone decreasing guarantees that ak − ak + 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}\rightarrow0 when m\rightarrow\infty, 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

\left|\sum_{k=0}^\infty(-1)^k\,a_k\,-\,\sum_{k=0}^m\,(-1)^k\,a_k\right|<a_{m+1}.

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

[edit] See also