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

Basic 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.

Eventually variation

The alternating series theorem can be generalized as follows: we do not require that the series be alternating or monotonically decreasing in magnitude right from the outset. Rather, we do require that the series eventually be alternating and eventually the terms be monotonically decreasing in magnitude.