In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit. The test was devised by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test.

For a generalization, see Dirichlet's test.

Leibniz discussed the criterion in his unpublished De quadratura arithmetica of 1676 and shared his result with Jakob Hermann in June 1705 and with Johann Bernoulli in October, 1713. It was only formally published in 1993.

A series of the form

$${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}=a_{0}-a_{1}+a_{2}-a_{3}+\cdots }$$

where either all a_n are positive or all a_n are negative, is called an alternating series.

The alternating series test guarantees that an alternating series converges if the following two conditions are met:

Moreover, let L denote the sum of the series, then the partial sum ${\textstyle S_{k}=\sum _{n=0}^{k}(-1)^{n}a_{n}\!}$ approximates L with error bounded by the next omitted term: