Evaluate the sum $\forall\:\:\alpha,\beta\in\mathbb{R}$ $$S=\sum_{n=1}^{\infty}\frac{\alpha^{n+1}-1}{n(n+1)}\sin\left(\frac{n\pi}{\beta}\right)$$
This is a follow up query to this question
The answer given there was partial and I couldn't understand that. The answer was
Easier: put $z=e^{\pm i\pi/\beta}$ and $z=\alpha e^{\pm i\pi/\beta}$ into $$\sum_{n=1}^\infty\frac{z^{n+1}}{n(n+1)}=z+(1-z)\log(1-z)\qquad(|z|\leqslant 1)$$
My question is what to do after this$?$ How to seperate the real and imaginary parts$?$ And how can a complex number inside a logarithmic function $?$
Any help is greatly appreciated.