If a series diverges, what happens when we divide by partial sums squared?

Question

I'm trying to prove or disprove:
It is given: $$a_n>0, \sum_{n=0}^\infty a_n=\infty$$
Prove / disprove: $$\sum_{n=0}^\infty \dfrac{a_n}{S_n^2} <\infty$$ where $$S_n=\sum_{k=1}^n a_k$$

I tried to prove using all the convergence tests I know, and tried to disprove using all kinds of sequences from $\frac{\ln(n)}{n}$ to $e^{n\ln(n)}$.

Thank you in advance.

Answer 1

Hint. Show that for $n\geq 1$, $$0<\frac{a_n}{S_n^2} \le \frac{1}{S_{n-1}}-\frac{1}{S_n}.$$ Then, by a telescopic argument, $$\sum_{n=0}^N \dfrac{a_n}{S_n^2} \leq \frac{1}{a_0}+\sum_{n=1}^N \left(\frac{1}{S_{n-1}}-\frac{1}{S_n}\right)=\frac{2}{a_0}-\frac{1}{S_N}.$$ What may we conclude?