I've been working on a recent exercise question where I was asked to show that: $$F(x)=\frac{x}{(1-x)^2}-\frac{2x}{(2-x)^2}=\sum_{n=0}^{\infty}n(1-2^{-n})x^n$$
Now I cansee that the infinite sum is a power series, which leads me to believe that I might argue that the Taylor series of a function is the power series for that function and then simply show that the taylor series for F(x) can be written as the sum seen on the far right. Now I've been able to show that the first term of the taylor series evaluated at x=0 is equivalent to the factor of the $x^1$ part of the infinite sum, but I'm confused as to how to go on from here.
I thought perhaps I might try to do some proof by induction, but that would require a generalized expression for the derivative of F(x) which doesn't exactly seem trivial.
I'd really appreciate some help as this has been puzzling me for hours.
If $|x|<1$, then$$\sum_{n=0}^\infty nx^n=\frac x{(1-x)^2}.$$Therefore, if $|x|<2$,$$\sum_{n=0}^\infty n\left(\frac x2\right)^n=\frac{x/2}{(1-x/2)^2}=2\frac x{(2-x)^2}.$$In other words,$$\sum_{n=0}^\infty n2^{-n}x^n=\frac{2x}{(1-x)^2}.$$Can you take it from here?