# Let $\{a_n\}$ be a sequence of positive numbers and $b_{n} = \frac{a_{n}}{(a_{1}+…+a_{n})^{2}}$. Prove $\sum_{n=1}^{\infty}b_{n}$ converges.

Let $$\{a_n\}_{n=1}^{\infty}$$ be a sequence of positive numbers and let $$b_{n} = \frac{a_{n}}{(a_{1}+...+a_{n})^{2}}$$ for n $$\in\mathbb{N}$$. Prove that $$\sum_{n=1}^{\infty}b_{n}$$ is a convergent series. I'm stuck on how to start this problem. I've considered the limit comparison test, but it hasn't worked out for me. I know I can assume $$\{a_{n}\}$$ and $$\{b_{n}\}$$ are positive, so maybe I need to show $$\{b_{n}\}$$ has an upper bound and apply the positive series test. Any help is appreciated. Thank you!

With $$S_n = a_1+ \ldots + a_n$$,
$$\frac{a_n}{S_n^2} \leqslant \frac{S_n - S_{n-1}}{S_nS_{n-1}} = \frac{1}{S_{n-1}}- \frac{1}{S_n}$$
Note that $$S_n$$ converges either to a finite limit or $$+\infty$$ and in all cases $$1/S_n$$ converges to a finite limit.