# Definition:Absolutely Convergent Series/Real Numbers

## Definition

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in the real number field $\R$.

Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is absolutely convergent if and only if:

$\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n}$ is convergent

where $\size {a_n}$ denotes the absolute value of $a_n$.