Consider the functional equation
$f(x)+f(\frac{x^2}{2})=x \mspace{10mu},\mspace{10mu}0<x<2$ .
$x\mapsto \frac{x^2}{2}$
$f(x)+f(\frac{x^2}{2})=x$
$f(\frac{x^2}{2})+f(\frac{x^4}{8})=\frac{x^2}{2}$
$f(\frac{x^4}{8})+f(\frac{x^8}{2^7})=\frac{x^4}{2^4}$ ...
$f(x)=x-\frac{x^2}{2}+\frac{x^4}{16}...=\sum_{k=0}^{\infty}(-1)^{(n+1)}(x/2)^{(2^n)} $
Can one get closed form of the series?