Definition:Absolutely Convergent Series

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

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

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

An absolutely convergent series is also convergent, as proved on Absolutely Convergent Series is Convergent.

Definition in a Number Field
Let $\displaystyle \sum_{n=1}^\infty a_n$ be a 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

 * Conditional convergence, the opposite of absolute convergence.
 * Absolutely Convergent Series is Convergent