I want to choose n balls from 2 types using generating functions.
Normally I would think to write $$f(x)=(1+x+...+x^n)^2 = \left ( \frac{1-x^{n+1}}{1-x} \right )^2$$ and then look for the coefficient of $x^n$, but I'm thinking that since any coefficient after $x^n$ won't contribute anything I should be able to use the simpler expression $$(1+x+...)^2 = \left ( \frac{1}{1-x} \right )^2$$ Is this correct? Is it something I would need to prove or is the simple explanation above sufficient?
It is correct that the simpler expression gives the same answer as the original. As to whether you need to prove it, that would depend on your audience. If you are a 1st-year undergraduate writing a homework assignment, the marker might want to be convinced that you know what you're doing. If you are writing a paper for Inventiones Mathematicae, you can safely assume the reader will fill in the dots.