I am trying to verify the results of a paper; to do so I have to compute the following summation:
$f(\theta)=\sum_{i=2}^\infty \frac{2i+1}{(i+2)(i-1)}P^2_i(\cos(\theta))$
as I am not a mathematician the first thing I tried to do was to approximate the summation with a finite number of terms, in the following figure I report $f(\theta)$ obtained with $1000$, $1001$, and $1002$ terms.
Looks like the series is not converged yet with $1000$ terms, and the same occurs if i employ $10000$ for which the computation begins to be cumbersome.
Edit: It is expected that the function $f(\theta)$ diverges when $\theta=0$ or $\theta=2 \pi$ however in the rest of the interval it should be finite.
So my question is: is it possible to obtain an analytical solution to the above summation? Perhaps employing some properties of the Legendre Polynomials (Generating functions, recursive relations). is the summation to infinity eventually convergent?
As I am clueless, any help is appreciated.