Absolutely convergent series

Definition

A series of real numbers is termed an absolutely convergent series if it satisfies the following equivalent conditions:

1. The series obtained by taking absolute values of all terms in the series is a convergent series. Explicitly, a series of the form $\sum _{k=1}^{\infty }a_{k}=a_{1}+a_{2}+a_{3}+\dots$ is absolutely convergent if $\sum _{k=1}^{\infty }|a_{k}|=|a_{1}|+|a_{2}|+|a_{3}|+\dots$ is a convergent series.
2. The sub-series of the series comprising the terms with positive values is convergent, and the sub-series of the series comprising the terms with negative values is also convergent.
3. Every rearrangement of the series is a convergent series and converges to the same value as the series itself.

Equivalence of definitions

For further information, refer: Riemann series rearrangement theorem