<s>[INST] Sum of series $\frac{x}{1-x^2} + \frac{x^2}{1-x^4} + \frac{x^4}{1-x^8} +\ldots$ 
The problem is to find the sum of series $$\frac{x}{1-x^2} + \frac{x^2}{1-x^4} + \frac{x^4}{1-x^8} +\ldots$$
  to infinite terms, if $|x|<1$ 

I tried taking $\frac{x}{1-x^2}$ outside but could not solve it.
How to proceed further?
 [/INST] Notice $$\frac{y}{1-y^2} = \frac{(1+y)-1}{1-y^2} = \frac{1}{1-y} - \frac{1}{1-y^2}$$
The sum at hand is a telescoping one. The partial sums has the form:
$$\sum_{n=1}^p
\frac{x^{2^{n-1}}}{1-x^{2^n}} = 
\sum_{n=1}^p
\left[\frac{1}{1-x^{2^{n-1}}} - \frac{1}{1-x^{2^n}}\right]
= \frac{1}{1-x} - \frac{1}{1-x^{2^p}}$$
When $|x| < 1$, $x^{2^p} \to 0$. This leads to
$$\sum_{n=1}^\infty
\frac{x^{2^{n-1}}}{1-x^{2^n}}
 = 
\lim_{p\to\infty}
\left[\frac{1}{1-x} - \frac{1}{1-x^{2^{p}}}\right]
= \frac{1}{1-x} - \frac{1}{1-0} = \frac{x}{1-x}$$
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