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

\sum_{n=1}^\infty a_n(-1)^n\!

where all the an are positive or 0, is called an alternating series. If the sequence an converges to 0, and each an is smaller than an-1 (i.e. the sequence an 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 \right \vert \le \left | S_k - S_{k-1} \right \vert = a_k\!

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:

\sum_{n=1}^\infty (1/3)^n = 1/2\!

[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