An alternating series is a series of the form

$$\sum_{i = 0}^{\infty}(-1)^ia_i$$

or

$$\sum_{i = 0}^{\infty}(-1)^{i+1}a_i$$

where $(a_n)$ is a non-negative sequence. Loosely, this is just a series where the terms ``alternate'' between positive and negative.

For convergence issues, see the following entry: alternating series test.