Absolutely convergent series

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 .

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 .

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.