How do I write this sum in summation notation? $$\sum_{n=?}^\infty \left(\frac{x^n}{?}\right) = \frac{x^0}{1} + \frac{x^1}{x^2 -1}+\frac{x^2}{x^4 - x^2 +1}+\frac{x^3}{x^6 -x^4 + x^2 -1}+\frac{x^4}{x^8-x^6 +x^4 - x^2 +1}+\cdots$$
I am pretty sure I have the numerator of the summation, $x^n$ correct, but don't know how to write the denominator because of the alternating signs in each term. As for the starting point, $n=?$, I think its going to be either $0$ or $1$, depending on the denominator - I don't want to be kicked from this site by attempting to divide by $0$ .

The denominators alternate signs, but they are of the form
$$\sum_{k=0}^m (-1)^k x^{2 k} = \frac{(-1)^{m+1} x^{2(m+1)}-1}{x^2+1}$$
The sum may then be written as
$$(x^2+1) \sum_{n=0}^{\infty} (-1)^n \frac{x^n}{(-1)^{n+1} x^{2(n+1)}-1} = -(x^2+1) \sum_{n=0}^{\infty} \frac{x^n}{x^{2(n+1)}+(-1)^n}$$
I should note that convergence is achieved when $|x|<1$ when $x$ is real.
