Alternating series test

The alternating series test is a method used to prove that infinite series of terms converge. It was discovered by Gottfried Leibniz and is sometimes known as Leibniz's test or Leibniz criterion.

A series of the form

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

where all the "a"_"n" are positive or 0, is called an alternating series. If the sequence "a"_"n" converges to 0, and each "a"_"n" is smaller than "a"_"n-1" (i.e. the sequence "a"_"n" is monotone decreasing), then the series converges. If "L" is the sum of the series,:$sum\_\{n=1\}^infty\; a\_n(-1)^n\; =\; L!$

then the partial sum

:$S\_k\; =\; sum\_\{n=1\}^k\; a\_n(-1)^n!$

approximates "L" with error

:$left\; |\; S\_k\; -\; L\; ight\; vert\; le\; left\; |\; S\_k\; -\; S\_\{k-1\}\; ight\; vert\; =\; a\_k!$

It is perfectly possible for a series to have its partial sums "S"_"k" fulfill this last condition without the series being alternating. For a straightforward example, consider:

:$sum\_\{n=1\}^infty\; left(frac\{1\}\{3\}\; ight)^n\; =\; frac\{1\}\{2\}!$

See also

*Dirichlet's test

References

* Knopp, Konrad, "Infinite Sequences and Series", Dover publications, Inc., New York, 1956. (§ 3.4) ISBN 0-486-60153-6

* Whittaker, E. T., and Watson, G. N., "A Course in Modern Analysis", fourth edition, Cambridge University Press, 1963. (§ 2.3) ISBN 0-521-58807-3

* Last, Philip, "Sequences and Series", New Science, Dublin, 1979. (§ 3.4) ISBN 0-286-53154-3