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Absolute Convergence

In Probability,

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

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

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

But even though n=1an\sum\limits_{n=1}^{\infty} a_n converges, it does not always imply it converges absolutely (conditional convergence). Example: {1,12,13,14,15}\{{-1}, {1 \over 2}, -{1 \over 3}, {1 \over 4}, -{1 \over 5}\}