Therefore, by the monotony criterion, the series\sum_{k=0}^\infty |x_k| is convergent if and only if the sequence (s_n)_{n\in\mathbb N} of its partial sums s_n:=\sum_{k=0}^n |x_k| is bounded.
By definition of absolutely convergent series, the series \sum_{k=0}^\infty x_k, is absolutely convergent if and only if the sequence (s_n)_{n\in\mathbb N} is bounded.