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


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


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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