Definitions

Alternating series

Alternating series

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 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}. 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,

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

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

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

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