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:

1. Alternating signs: All the $a_{k}$s are nonzero and the sign of $a_{k+1}$ is opposite the sign of $a_{k}$.
2. Monotonically decreasing in magnitude: $|a_{k}|\geq |a_{k+1}|$ for all $k$.
3. 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.