Consider the infinite summation : $$\sum_{n=1}^{\infty}n\left[1-\ _{2}F_{1}\left(1,-\frac{s}{n};1-\frac{s}{n};z^{n} \right ) \right ]$$ Where $_{2}F_{1}\left(a,b;c;z\right)$ is the Gauss' hypergeometric function, $|z|<1$, and $s \in \mathbb{C},\Re(s)>0$. This summation seems to converge except when $s$ is a positive integer. But i don't know how to prove it. By the same token, is there a closed form for this sum ?


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