Sum of the series $\frac{1}{(1-x)(1-x^3)}+\frac{x^2}{(1-x^3)(1-x^5)}+\frac{x^4}{(1-x^5)(1-x^7)}+....$ If $|x|<1$, find the sum of infinite terms of following series:
$$\frac{1}{(1-x)(1-x^3)}+\frac{x^2}{(1-x^3)(1-x^5)}+\frac{x^4}{(1-x^5)(1-x^7)}+....$$
Could someone give me hint to solve this problem. I wrote $r_{th}$ term of the series but can't find sum of $n$ terms. If somehow I can get sum of $n$ term then I can put $n \to \infty $ to get sum of infinite terms.

$\frac{1}{(1-x)(1-x^3)}=\frac{1}{x-x^3}(\frac{1}{1-x}-\frac{1}{1-x^3})$
$\frac{x^2}{(1-x)(1-x^5)}=\frac{1}{x-x^3}(\frac{1}{1-x^3}-\frac{1}{1-x^5})$
...
So, assuming n starts from 1, $\frac{1}{(1-x)(1-x^3)}+\frac{x^2}{(1-x^3)(1-x^5)}+\frac{x^4}{(1-x^5)(1-x^7)}...=\frac{1}{x-x^3}(\frac{1}{1-x}-\frac{1}{1-x^{2n+1}})$
