Find the value of $\lim_{n\to \infty}\sum_{k=0}^n\frac{x^{2^k}}{1-x^{2^{k+1}}}$.

Question

If $0 \lt x \lt 1$ and

$$A_n=\frac{x}{1-x^2}+\frac{x^2}{1-x^4}+.....+\frac{x^{2^n}}{1-x^{2^{n+1}}}$$ then

Find $\lim\limits_{n\to \infty}A_n$.

Answer 1

Expand each term into geometric series. Then in the limit $n\rightarrow \infty$ the sum becomes $$\sum_{k=1}^{\infty}x^k=\frac{x}{1-x}.$$ The same answer can be obtained more formally by subtracting $\frac{x}{1-x}$ from the initial series and observing that the result telescopes to $$-\frac{x^{2^{n+1}}}{1-x^{2^{n+1}}}.$$