algebraic manipulation of probability generating functions

I am revising for a statistics test and I have an urgent problem involving probability generating functions. I understand the concept its just the algebraic manipulation that gets me. My exam is only a few days away so if you could help me out that would be GREAT! :)

I'm supposed to determine the probability distribution of X from the pgf below.

Here is the working I have done so far:

for $\displaystyle (0<\alpha<1)$,

$\displaystyle \pi(z)=\frac{1-\alpha(1-z)}{1+\alpha(1-z)} = \frac{\frac{1-\alpha}{1+\alpha}+\frac{\alpha}{1+\alpha}z}{1-\frac{\alpha}{1+\alpha}z} $

$\displaystyle = (\frac{1-\alpha}{1+\alpha}+\frac{\alpha}{1+\alpha}z)(1+[\frac{\alpha}{1+\alpha}z] +[\frac{\alpha}{1+\alpha}z]^2+...) $

therefore $\displaystyle P(X=0)=\frac{1-\alpha}{1+\alpha}$

This matches exactly with what the lecturer had done exept I couldn't understand how he leapt to this next part:

*and* for x=>1, $\displaystyle P(X=x)=(\frac{1-\alpha}{1+\alpha})(\frac{\alpha}{1+\alpha})^x + (\frac{\alpha}{1+\alpha})(\frac{\alpha}{1+\alpha}) ^{x-1}$

$\displaystyle =(\frac{\alpha}{1+\alpha})^x(\frac{1-\alpha}{1+\alpha}+1)=\frac{2\alpha^x}{(1+\alpha)^{ x+1}}$

Could someone please explain to me how he got this?