Finding power series representation of the fucntion $f(x) = \frac{1+x^2}{1-x^2}$. So I want to find the power series representation of the function $f(x) = \frac{1+x^2}{1-x^2}. $ So how do I solve this?
I can start by reducing the function to the known form of geometric series but I have a summation at the numerator. How do I overcome that?
$\frac{1+x^2}{1-x^2} = \frac{(1+x)^2 -2x}{1-x^2}  = \frac{1+x}{1-x} - \frac{2x }{1-x^2}$, now second part I can solve, it would be $2x \sum_{n =0}^{\infty} x^{2n}.$ How do I go about the first part? Please give some hint. Thank You.

I would go straight using with $X=x^2$
$${1\over 1-X}=\sum_{n=0}^\infty X^n$$
This leads to
$${1+x^2\over 1-x^2}=\left(1+x^2\right)\sum_{n=0}^\infty x^{2n}$$
Now rearranging the sum
$${1+x^2\over 1-x^2}=1+2\sum_{n=1}^\infty x^{2n}$$
