Hello guys i have a problem evaluating the following sum $$\sum_{n=1}^{+\infty}\frac{n(n+1)}{2}\frac{4x(3\pi ^{2}(n+1)^{2}+x^{2})}{(x^{2}-\pi ^{2}(n+1)^{2})^{3}}$$ It is obviously of the polygamma form but i cant find an idea to solve it. Writing it into partial fractions makes it diverge so i dont know where to go. Please give a hint or a solution, anything is helful. If we were to write it into part frac we would get $$\frac{(n^2+n)}{(x-\pi (n+1))^{3}}$$ which is similar to $\frac{1}{n}$ which diverges, therefore i need an idea that doesnt rely on it.
2026-03-25 20:40:08.1774471208
Polygamma sum problem ...
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