I have this following series: \begin{equation} G = \frac{-1}{4\pi}\sum_{l=0}^{\infty}\frac{2l+1}{\frac{l(l+1)}{R^2}+\frac{1}{\alpha}}P_l(\cos(\gamma)) \end{equation} where $R= 6371$, $\alpha>0$ fixed.
I want to check for which values of $\alpha$ this series converges. Can anyone help me where to start since I have little knowledge of this kind of series? ( I don't mind evaluating the series numerically using some code ... ).
We have to study $$f\left(w,\gamma\right)=\sum_{l\geq0}\frac{\left(2l+1\right)P_{l}\left(\cos\left(\gamma\right)\right)}{l\left(l+1\right)+w},\,w\in\mathbb{R}^{+}.$$ If $\cos\left(\gamma\right)=1$, since $P_{l}\left(1\right)=1,\ \forall l\in\mathbb{N}$ we have $$f\left(w,1\right)=\sum_{l\geq0}\frac{2l+1}{l\left(l+1\right)+w}\sim\sum_{l\geq0}\frac{1}{l+w}$$so the series diverges like the harmonic series. So assume that $\cos\left(\gamma\right)\neq1$. We have $$f\left(w,\gamma\right)=2\sum_{l\geq0}\frac{lP_{l}\left(\cos\left(\gamma\right)\right)}{l\left(l+1\right)+w}+\sum_{l\geq0}\frac{P_{l}\left(\cos\left(\gamma\right)\right)}{l\left(l+1\right)+w}\tag{1}$$ the second series of $(1)$ is convergent since $$\sum_{l\geq0}\left|\frac{P_{l}\left(\cos\left(\gamma\right)\right)}{l\left(l+1\right)+w}\right|\leq\sum_{l\geq0}\frac{1}{l^{2}+w}<\infty,\,\forall w\in\mathbb{R}^{+}$$ since $$\sum_{l\geq0}\frac{1}{l^{2}+w}=\frac{1}{w}+\sum_{l\geq1}\frac{1}{l^{2}+w}\leq\frac{1}{w}+\sum_{l\geq1}\frac{1}{l^{2}}$$ by the $p$-test. so we have to study $\sum_{l\geq0}\frac{lP_{l}\left(\cos\left(\gamma\right)\right)}{l\left(l+1\right)+w}$. We have that $$\sum_{l\geq0}\frac{lP_{l}\left(\cos\left(\gamma\right)\right)}{l\left(l+1\right)+w}\leq\sum_{l\geq0}\frac{P_{l}\left(\cos\left(\gamma\right)\right)}{l+1}=\sum_{l\geq0}P_{l}\left(\cos\left(\gamma\right)\right)\int_{0}^{1}t^{l}dt$$ and from the generating function of the Legendre polynomials we get $$\sum_{l\geq0}\frac{P_{l}\left(\cos\left(\gamma\right)\right)}{l+1}=\int_{0}^{1}\frac{1}{\sqrt{1-2\cos\left(\gamma\right)t+t^{2}}}dt\tag{2}$$ and the integral in $(2)$ is convergent since $$\int_{0}^{1}\frac{1}{\sqrt{1-2\cos\left(\gamma\right)t+t^{2}}}dt=\log\left(2+2\sqrt{2-\cos\left(\gamma\right)}-\cos\left(\gamma\right)\right)-\log\left(2-\cos\left(\gamma\right)\right)$$ so the series is convergent.