 Solution

Suppose ${\sum }_n{a}_n$ diverges. Positivity implies $s_n\to \infty$. Put $t_n=a_n/s_n\in (0,1)$ for $n\ge 2$. Since $s_{n-1}=s_n-a_n$,If ${\sum }_n{t}_n$ converged, then eventually $t_n\le 1/2$ andSumming would bound the telescoping quantity $\log (s_n/s_1)$, contradicting $s_n\to \infty$. ThereforeThe converse is true. If ${\sum }_n{a}_n<\infty$, then $s_n\ge s_1>0$ andTaking the contrapositive shows that divergence of the ratio series forces divergence of ${\sum }_n{a}_n$.