Checking the boundedness of $A_n = \frac{x}{1-x^2} + \frac{x^2}{1-x^4} + \ldots + \frac{x^{2^n}}{1-x^{2^{n+1}}}$ Sequence is given by $$A_n = \frac{x}{1-x^2} + \frac{x^2}{1-x^4} + \ldots + \frac{x^{2^n}}{1-x^{2^{n+1}}}.$$ Please advise me how to show that this sequence is bounded above by 1. 0

\begin{align*}
  \frac{x^{2^{n}}}{1-x^{2^{n+1}}}+\color{red}{\frac{1}{1-x^{2^{n+1}}}} &=
  \frac{1}{1-x^{2^{n}}}  \\
  \frac{x^{2^{n-1}}}{1-x^{2^{n}}}+\frac{1}{1-x^{2^{n}}} &=
  \frac{1}{1-x^{2^{n-1}}}  \\
  & \: \; \vdots \\
  \frac{x}{1-x^{2}}+\frac{1}{1-x^{2}} &=
  \color{blue}{\frac{1}{1-x}}  \\
  A_{n} &= \color{blue}{\frac{1}{1-x}}-
           \color{red}{\frac{1}{1-x^{2^{n+1}}}} \\
  \lim_{n\to \infty} A_{n} &=
  \left \{
    \begin{array}{ccc}
      \displaystyle \frac{x}{1-x} & \text{for} & -1<x<1 \\
      \displaystyle \frac{1}{1-x}  & \text{for} &  x^2 > 1 \\
    \end{array}
  \right.
\end{align*}
