Definition:Absolutely Convergent Series

General Definition
Let $V$ be a normed vector space with norm $\left\Vert{\cdot}\right\Vert$.

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$.

Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ in $V$ is said to be absolutely convergent iff $\displaystyle \sum_{n \mathop = 1}^\infty \left\Vert{a_n}\right\Vert$ is a convergent series in $\R$.

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

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

Also see

 * Definition:Conditional Convergence, the antithesis of absolute convergence.
 * Absolutely Convergent Series is Convergent, which shows that an absolutely convergent series in a Banach space is also convergent.