Let $\{a_n\}$ be a sequence of positive numbers. Prove that the convergence of $$\sum_{n=1}^\infty \frac{1}{a_n}$$ implies the convergence of $$\sum_{n=1}^\infty \bigg(n^2a_n\bigg(\sum_{k=1}^n a_k\bigg)^{-2}\bigg).$$
Actually I have no idea about where to start this. I tried few of the relevant series convergence tests, but none of them worked. Any detailed help will be appreciated.
