If $a_n>0$ so that $ \sum_n \frac{a_n}{s_n}$ diverges, $s_n = \sum_{i=1}^n a_i$, must $ \sum_n a_n$diverge?

Question

I have seen questions which make you prove the converse of this which is true. I am not sure however, whether this statement itself is true?

Answer 1

Yes, $\sum a_n$ diverges by the Cauchy criterion.

First, notice that $$ \frac{a_{n+1}}{s_{n+1}}+\dotsb+\frac{a_{n+p}}{s_{n+p}} \le \frac{s_{n+p}-s_n}{s_{n+1}}, $$ and since the sum in the left-hand side tends to infinity for any fixed value of $p$, so does the expression in the right-hand side. But the denominators $s_{n+1}$ are bounded away from $0$; hence $s_{n+p}-s_n\to\infty$ for any fixed $p$, showing that $\sum a_n$ diverges.