Argue for that the sum function $F$ for a power series satifies that $F(x) = \frac{x}{(1-x)^2} - \frac{2x}{(2-x)^2}$ when $-1<x<1$

Question

Consider the power series $$ \sum_{n=0}^\infty n(1-2^{-n})z^n $$ Then I have to argue for that the sum function $F$ for the given power series satifies that $$ F(x) = \frac{x}{(1-x)^2} - \frac{2x}{(2-x)^2} $$ when $-1<x<1$. To be honest I have no idea how to even start this question. I have looked in my book if I could find any sentence that would help me further but I couldn't. Can you help me?

Thanks in advance.

Answer 1

$$\sum_{n=0}^\infty n(1-2^{-n})z^n=\sum_{n=0}^\infty nz^n-\sum_{n=0}^\infty n\left(\frac{z}{2}\right)^n$$ $$\sum_{n=0}^\infty nz^n=z\sum_{n=0}^\infty nz^{n-1}=z \left(\sum_{n=0}^\infty z^{n} \right)'$$