We have the M-series for $\frac{1}{1-x} = \sum_{n=0}^\infty x^n, \frac{1}{1+x} = \sum_{n=0}^\infty (-x)^n,$ and $\frac{1}{1-x^2} = \sum_{n=0}^\infty (x^2)^n$. I need to use the product of the first two convergent series to find the M-series of the last.
That is, find the product $\sum_{n=0}^\infty x^n \cdot \sum_{n=0}^\infty (-x)^n = \sum_{n=0}^\infty \cdot \sum_{k=0}^n x^k \cdot (-x)^{n-k}$. This is homework; hints would be appreciated.
Try using the first result in here: http://en.wikipedia.org/wiki/Cauchy_product
After that, you will have to notice that the coefficients $a_k$ and $b_k$ are either +1 or -1 and, because of that, the coefficients of $x^{2n+1}$ equal zero, while the coefficients of $x^{2n}$ equal one. Thus, the sum can be written like that.
If you have trouble in finding the coefficients of $x^{2n}$ and $x^{2n+1}$, then I suggest working through the first few initial cases. ok?