Let $\displaystyle a_n$ be positive, and $\displaystyle \sum_{n=1}^\infty a_n$ be divergent. Assume $\displaystyle S_n=a_1+\cdots+a_n$. Prove that for any $\displaystyle \sigma>1,$ the series $\displaystyle \sum_{n=1}^\infty \frac{a_n}{S_n^\sigma}$ converges.