Can someone help me finish this: evaluate $S_n = \frac{x}{1-x^2}+\frac{x^2}{1-x^4}+ ... + \frac{x^{2^{n-1}}}{1-x^{2^{n}}}$ I am asked to find the closed form solution for the below.
$$S_n = \frac{x}{1-x^2}+\frac{x^2}{1-x^4}+ ... + \frac{x^{2^{n-1}}}{1-x^{2^{n}}}$$
Just writing out the $S_1, S_2, S_3$, I have managed to find a pattern, which is:
$$S_n = \frac{S_{n-1}}{1-x^{2^n}} + \frac{x^{2^n}}{1-x^{2^{n+1}}}$$
I am not sure how to proceed onwards to solve this recurrence relation. Is there a clever trick I can do to solve it?

$$\begin{align} \frac{x}{1-x}-S_n &= \\
&= \frac{x}{1-x}-\frac{x}{1-x^2}-\frac{x^2}{1-x^4}- \ldots - \frac{x^{2^{n-1}}}{1-x^{2^{n}}} \\
&=\frac{x^2}{1-x^2}-\frac{x^2}{1-x^4}- \ldots - \frac{x^{2^{n-1}}}{1-x^{2^{n}}} \\
&=\ldots \\
&=\frac{x^{2^{n-1}}}{1-x^{2^{n-1}}}- \frac{x^{2^{n-1}}}{1-x^{2^{n}}} \\
&=\frac{x^{2^{n}}}{1-x^{2^{n}}}. \end{align}$$
So,
$$S_n= \frac{x}{1-x}-\frac{x^{2^{n}}}{1-x^{2^{n}}}.$$
