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 $$a_n tends to zero and is monotone decreasing (at least from a certain point on), it can be easily shown that the sequence of partial sums is a Cauchy sequence. Assuming $$m,

$$egin{array}{rcl}displaystyleleft|sum_{k=0}^m(-1)^k,a_k,-,sum_{k=0}^n,(-1)^k,a_k ight|&=&displaystyleleft|sum_{k=m+1}^n,(-1)^k,a_k ight|=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

"(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

$$left|sum_{k=0}^infty(-1)^k,a_k,-,sum_{k=0}^m,(-1)^k,a_k ight|

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