Alternating series theorem

This article describes a test that is used to determine, in some cases, whether a given infinite series or improper integral converges. It may help determine whether we have absolute convergence, conditional convergence, or neither.
View a complete list of convergence tests

Statement

Consider a series of the form:

$a_{1} + a_{2} + \dots + a_{n} + \dots = \sum_{k = 1}^{\infty} a_{k}$

Suppose the following three conditions hold for the series:

Alternating signs: All the $a_{k}$ s are nonzero and the sign of $a_{k + 1}$ is opposite the sign of $a_{k}$ .
Monotonically decreasing in magnitude: $| a_{k} | \geq | a_{k + 1} |$ for all $k$ .
Terms approach zero: $lim_{k \to \infty} a_{k} = 0$ .

Then the series is a convergent series. It may be an absolutely convergent series or a conditionally convergent series, depending on whether the series of the absolute values of its terms converges.