I came across an electrostatic problem which deals with $\sum_{l=0}^{\infty} (P_{l}(0))^2$. Does this series converge? If yes, how can we calculate it?
(By completeness theorem, I know that $\sum_{l=0}^{\infty} (2l + 1)(P_{l}(0))^2 = 2$, but I am not able to simplify further.)