Convergence of the series $ \frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+\frac{8x^7}{1+x^8}+...$ The series $ \frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+\frac{8x^7}{1+x^8}+...$
(A) is uniformly convergent for all x
(B) is convergent for all x but the convergence is not uniform
(C) is convergent only for $|x|\le \frac{1}{2}$ but the convergence is not uniform
(D) is uniformly convergent on $[-\frac{1}{2},\frac{1}{2}]$
My approach is if we take $ f(x)=\frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+\frac{8x^7}{1+x^8}+...$
then $$ \int f(x)dx=\log((1+x^2)(1+x^4)(1+x^8)...)=\log\left(\frac{1-x^{2^n}}{1-x^2}\right)$$
Hereafter, I am stuck.

The answer is D). For $|x| \leq \frac1  2 $ we have $|\frac {n x^{2n-1}} {1+x^{2n}}| \leq  n|x|^{2n-1} $ and $ \sum n |\frac 1 2 |^{n}$ is convergent.
