I want to integrate the following amazing integral with Legendre Polynomials. If you need it for your solution, it might be good to know, that the series converges absolutely. I do not really have an idea:
$$\int_0^{\pi} \left(\sum_{l=0}^{\infty} f(l) P_l(\cos(\theta))\right)^2sin(\theta) d \theta $$
Make the change of variable $u = \cos(\theta)$ then you get
$$\int_{-1}^1\sum_{l=0}^{\infty}f(l)P_l(u)\sum_{k=0}^{\infty}f(k)P_k(u)du = \int_{-1}^1\sum_{l=0}^{\infty}\sum_{j=0}^lf(l)f(j-l)P_l(u)P_{k-l}(u)du$$
From there, make use of what I hope is uniform convergence of your series and interchange the sum and integral and make use of orthogonality of Legendre polynomials.