<s>[INST] $f(x)=\sum_{i=0}^{\infty} (x^{2^n})/(1-x^{2^{n+1}})$. Find $f(99)$. $f(x)=\sum_{i=0}^{\infty} (x^{2^n})/(1-x^{2^{n+1}})$. Find $f(99)$.
ATTEMPT: The following series can be re-written as $f(x)=\sum_{i=0}^\infty \left(\frac{1}{1-x^{2^n}}\right) \cdot \left( \frac{1}{1+x^{2^n}}\right)$ and then expanded along.
 [/INST] If this problem is $$f(x)=\sum_{n=0}^{\infty}\dfrac{x^{2^n}}{1-x^{2^{n+1}}}$$
Hint:Let $x^{2^n}=t$,then
$$\dfrac{x^{2^n}}{1-x^{2^{n+1}}}=\dfrac{t}{1-t^2}=\dfrac{1}{1-t}-\dfrac{1}{1-t^2}=\dfrac{1}{1-x^{2^n}}-\dfrac{1}{1-x^{2^{n+1}}}$$
so
$$f(x)=\dfrac{1}{1-x}-\lim_{N\to\infty}\dfrac{1}{1-x^{2^{N+1}}}=\dfrac{1}{1-x}$$
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