Alternating series test
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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
where all the an are positive or 0, is called an alternating series. The sequence an converges to 0 if each an is smaller than an-1 (monotone decreasing); if that is so, then the series converges, say to L
and the partial sum
approximates the sum L of the series with error
It is perfectly possible for a series to have its partial sums Sk fulfill this last condition without the series being alternating. For a straightforward example, consider:
[edit] 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