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Consider the following series (where $l,\,m\in\mathrm{Z}\,$):

$S = \displaystyle\sum^{\infty}_{l\,=\,2} \frac{2l+1}{(l-1)(l+2)(1+l^2)}\sum^{l}_{m\,=\,-l}\frac{(l-m)!}{(l+m)!}\Big(P^m_l(x)\,\Big)^2$,

where $P^m_l(x)$ is the associated Legendre polynomials and $x\in[\,0,\,1]$.

I have some difficulty to show that $S$ converges. Can someone help me?

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