Alternating series theorem

Statement

Consider a series of the form:

$a_1+a_2+\dots +a_n+\dots =\sum _{k=1}^{\mathrm{\infty }}{a}_k$

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: $\left|a_k\right|\ge \left|a_{k+1}\right|$ for all $k$.
3. Terms approach zero: $\underset{k\to \mathrm{\infty }}{lim}{a}_k=0$.

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.