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.

Hint: $$\frac{x^{2n}}{(1-x^{2n+1})(1-x^{2n+3})}=\frac{1}{x(1-x^2)}\left(\frac{1}{1-x^{2n+1}}-\frac{1}{1-x^{2n+3}}\right)$$
