Absolute Convergence

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In Probability,

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

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n = 1 \sum \infty ∣ a_{n} ∣

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: $\{{-1}, {1 \over 2}, -{1 \over 3}, {1 \over 4}, -{1 \over 5}\}$

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