Absolutely convergent series

A series $\sum_{n=1}^\infty a_n$ with $a_n\in\mathbb{R}$ or $\mathbb{C}$ is absolutely convergent if the series of absolute values converges :

Absolute convergence is stronger than ordinary convergence and has powerful consequences: in $\mathbb{R}$ or $\mathbb{C}$ it implies convergence of $\sum a_n$ and stability under rearrangement .

Examples:

• $\sum_{n=1}^\infty \frac{1}{n^2}$ is absolutely convergent since $\sum |1/n^2|$ converges.
• $\sum_{n=1}^\infty (-1)^n\frac{1}{n^2}$ is absolutely convergent.
• The series $\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}$ is not absolutely convergent (its absolute value series is harmonic and diverges).