Absolute Convergence

In Probability,

$\sum\limits_{n=1}^{\infty} a_n$ converges absolutely if and only if $\sum\limits_{n=1}^{\infty} |a_n|$ converges.

Because $- \sum\limits_{n=1}^{\infty} |a_n| \leq \sum\limits_{n=1}^{\infty} a_n \leq \sum\limits_{n=1}^{\infty} |a_n|$,

If $\sum\limits_{n=1}^{\infty} a_n$ converges absolutely, then it converges.

But even though $\sum\limits_{n=1}^{\infty} a_n$ converges, it does not always imply it converges absolutely (conditional convergence). Example: $\{{-1}, {1 \over 2}, -{1 \over 3}, {1 \over 4}, -{1 \over 5}\}$