I have to show that this dreadfull thing is infinite $\forall a\in(-1,1)\backslash\{0\}$ $$f(a)=\sum_{n=0}^\infty \left(\frac{1-(\pm a/2)^{n+1}}{(2\mp a) a^{n+1}}\right)^2$$
I can show that if $0<a_1>a_2<1$ then $f(a_2)<f(a_1)$ and conversely for $-1<a_2<a_1<0$ but I don't know how to get further than that :(