<s>[INST] Generating function expansion I am working on a generating function for a sequence problem and I am stuck with expanding.
I have 
$$\frac{1}{4(1-x)^2} - \frac{1}{4(x+1)^2} $$
From my notes, the first term will be $\sum_{n=0}^{\infty}\frac{1}{4}(n+1)x^n.$
I am stuck on the second term where the denominator is $(x+1)^2$  Am I able to make it look like $(1-(-x))^2$ and do something similar to the first term and get
$$\sum_{n=0}^{\infty}\frac{1}{4}(-1)^n(n+1)x^n
$$
 [/INST] You can observe 
$\frac{1}{4(1-x)^2} - \frac{1}{4(x+1)^2}=\frac{x}{(1-x^2)^2}=\frac{x}{1-x^2}\sum_{n=0}^\infty x^{2n}$
$=\sum_{n=0}^\infty \frac{x^{2n+1}}{1-x^2}$
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