Test the uniform convergence of the series
$$\sum_{n=1}^\infty \frac{2^nx^{2^n-1}}{1+x^{2n}} $$
I have tried the tests listed in my book as well as the definitional method. Nothing seems to work. Need help/hint.
My work:
Using partial sum method
$S_n=\sum_{n=1}^n \frac{2^nx^{2^n-1}}{1+x^{2n}}= 1+\frac{2}{1+x^2}+\frac{4x^3}{1+x^4}+\frac{8x^7}{1+x^6}+\frac{16x^{15}}{1+x^8}+...+\frac{2^nx^{2^n-1}}{1+x^{2n}} $
How can I proceed from here?