Alternating series test

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


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