Putnam 1966 B3

B-3. Show that if the series $$\sum_{n=1}^{\infty} \frac{1}{p_{n}}$$ is convergent, where $p_{1}, p_{2}, p_{3}, \cdots, p_{n}, \cdots$ are positive real numbers, then the series $$\sum_{n=1}^{\infty} \frac{n^{2}}{\left(p_{1}+p_{2}+\cdots+p_{n}\right)^{2}} p_{n}$$ is also convergent.