Evaluate in closed form: $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1-x^{2^{n+1}}}$

Question

Evaluate in closed form: $$\sum_{n=0}^{\infty} \frac{x^{2^n}}{1-x^{2^{n+1}}}$$

where $|x|<1$ I am stuck on this problem. I tried decomposing the denominator into a geometric sum and tried to use binary representations of integers but it seems to be a dead end.

I would appreciate any hints. I prefer hints to complete solutions.

Answer 1

Hint:

$$\frac{x^{2^n}}{1-x^{2^{n+1}}}=\frac{1}{1-x^{2^n}}-\frac{1}{1-x^{2^{n+1}}}$$ for all $\;n\in\mathbb{N}\cup\{0\}.$

Therefore:

$$\sum_{n=0}^{N} \frac{x^{2^n}}{1-x^{2^{n+1}}}=\frac{1}{1-x}-\frac{1}{1-x^{2^{N+1}}}$$