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.
