Partial sum $\{\frac{s_m}{s_n}\}$ converge to $1$ implies series converge?

Question

Let $s_n$ and $s_m$ be the partial sum of the series $\sum\limits_{k=0}^\infty a_k$ with $m<n$ and $a_k > 0$ for all k. If $\{\frac{s_m}{s_n}\}$ converge to $1$, does it imply that the series converges?

Answer 1

If $\frac {s_m} {s_n} \to 1$ as $n,m \to \infty$ then $\ln s_m -\ln s_n \to 0$ which means $(\ln s_n)$ is a Cauchy sequence. Hence it converges to some number $c$. It follows that $s_n \to e^{c}$ so the series is convergent.