Definition:Absolutely Convergent Series

Definition
Let $\displaystyle \sum_{n=1}^\infty a_n$ be a convergent series in one of the standard number fields $\Q, \R, \C$.

Then $\displaystyle \sum_{n=1}^\infty a_n$ is said to be absolutely convergent iff $\displaystyle \sum_{n=1}^\infty \left|{a_n}\right|$ is convergent.

Also see
A series which is convergent, but not absolutely convergent, is said to be conditionally convergent.