Show that the following limit does not exist: $$\sum_{0}^{\infty} (-1)^n x^{(2^n)}\text{ with }x \uparrow 1$$
I tried setting $$f(x) = x - x^2 + x^4 - x^8...$$ then $$f(x) = x-f(x^2)$$ then the limit if it exists should be $$l=1-l\text{ so that }l=\frac{1}{2}$$
Also $$f(x)= x-x^2+f(x^4)$$ But after that I am stuck...