Absolutely convergent series - Imported Knowlpedia Content

An absolutely convergent series is a series $\sum_{n=1}^\infty a_n$ such that the series of absolute values $\sum_{n=1}^\infty |a_n|$ is a convergent series.

Examples

$\sum_{n=1}^\infty \frac{1}{n^2}$ is absolutely convergent.
$\sum_{n=1}^\infty \frac{(-1)^n}{n^2}$ is absolutely convergent, since $\sum_{n=1}^\infty \left|\frac{(-1)^n}{n^2}\right|=\sum_{n=1}^\infty \frac{1}{n^2}$ converges.

Remarks

Absolute convergence is stronger than ordinary convergence: see absolute convergence implies convergence. It also ensures stability under rearrangements, formalized by the rearrangement theorem for absolutely convergent series.