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. In mathematics, a series is often represented as the sum of a sequence of terms. ... In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ... Leibniz redirects here. ...

A series of the form

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, In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ... In mathematics, an alternating series is an infinite series of the form with an ≥ 0. ... For other senses of this word, see sequence (disambiguation). ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...

then the partial sum

approximates L with error

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:

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