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In mathematics, an alternating series is an infinite series of the form## See also

- $sum\_\{n=0\}^infty\; (-1)^n,a\_n,$

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

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 $mmath>,$

$begin\{array\}\{rcl\}\; displaystyleleft|sum\_\{k=0\}^m(-1)^k,a\_k,-,sum\_\{k=0\}^n,(-1)^k,a\_kright|\&=\&displaystyleleft|sum\_\{k=m+1\}^n,(-1)^k,a\_kright|=a\_\{m+1\}-a\_\{m+2\}+a\_\{m+3\}-a\_\{m+4\}+cdots+a\_n\; \&=\&displaystyle\; a\_\{m+1\}-(a\_\{m+2\}-a\_\{m+3\})\; -\; (a\_\{m+4\}-a\_\{m+5\})\; -cdots-a\_n\{m+1\}\; end\{array\}\; math>$

(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

$left|sum\_\{k=0\}^infty(-1)^k,a\_k,-,sum\_\{k=0\}^m,(-1)^k,a\_kright|\{m+1\}.\; math>$

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

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Last updated on Saturday August 30, 2008 at 08:43:22 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday August 30, 2008 at 08:43:22 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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