Let $S(n)=\sum_{k=1}^{n}k^2$, what is the value of $\sum_{i=1}^{\infty} \frac{1}{S(i)}$ ?
The expression of $S(n)$ is obvious. The calculus of something like $\sum_{i=1}^{\infty} 1/k^2$ involves some classical Fourier series. I was wondering if we could come up with useful ideas in order to solve the question on the first line. Stuck. Any hint ?