Remarks

1. The harmonic series $\sum_{k=1}^\infty 1/k$ shows that the implication can not be reversed.
2. This result can be used as a first test for convergence of a series $\sum_{k=1}^\infty a_k$ . If $a_k$ does not converge to $0$ , then $\sum_{k=1}^\infty a_k$ does not converge either.

Proof. Let be the value of the sum, and let $\varepsilon>0$ be arbitrary. Then there exists an $N\ge 1$ such that $$ | \sum_{k=1}^M a_k -S | < \frac{\varepsilon}{2} $$ for all $M\ge N$ . For $j\ge N$ we then have \begin{eqnarray*} |a_{j+1}| &=& | \sum_{k=1}^{j+1} a_k -\sum_{k=1}^j a_k| \\ &\le & | \sum_{k=1}^{j+1} a_k -S | + |S - \sum_{k=1}^j a_k| \\ &<& \varepsilon, \end{eqnarray*}and the claim follows.