You know that $\displaystyle \frac{1}{1-t} = \sum_{n=0}^{\infty}t^n$. If $\displaystyle t = x^2$, then $\displaystyle \frac{1}{1-x^2} = \sum_{n=0}^{\infty}(x^2)^n = \sum_{n=0}^{\infty}{x}^{2n}$.
You know that $\displaystyle \frac{1}{1-t} = \sum_{n=0}^{\infty}t^n$. If $\displaystyle t = x^2$, then $\displaystyle \frac{1}{1-x^2} = \sum_{n=0}^{\infty}(x^2)^n = \sum_{n=0}^{\infty}{x}^{2n}$.
I proved it about an hour ago. Thanks anyway!
Cheers!