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 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 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 $m < n$ ,

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

(the sequence being monotone decreasing guarantees that $a k - a k + 1 > 0$ ; 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 $m\rightarrow\infty$ , 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

Nörlund-Rice integral

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