TITLE: Prove the series $\sum_{n=1}^{\infty} \frac{n^2{a_n}}{(a_1 + a_2 + ... + a_n)^{2}}$ converges
QUESTION [1 upvotes]: Prove the series $\sum_{n=1}^{\infty} \frac{n^2{a_n}}{(a_1 + a_2 + \dots + a_n)^{2}}$  converges given $\sum_{n=1}^{\infty} \frac{1}{a_n}$ convereges, $\forall a_n > 0$
This is a question that a friend of mine has studied throughout the last semester and failed to solve. I think he would like to figure out this question but needs some hints.

REPLY [2 votes]: Define $B_n = \dfrac{n^2a_n}{S^2_n}, S_n =\sum_{k=1}^n a_k, R_n= \sum_{k=1}^n \dfrac{1}{a_k}$. By Cauchy inequality twice: $S_n \ge \dfrac{n^2}{R_n}$. So $R_n  \ge \dfrac{n^2}{S_n}$. Thus $B_n = \dfrac{n^2}{S_n}\cdot \dfrac{a_n}{S_n}\le \dfrac{a_n}{S_n}\cdot R_n\le R_n$. Thus by comparison test, $\sum B_n$ converges !
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