Convergence of the series $ \frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+\frac{8x^7}{1+x^8}+...$

Question

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.

Answer 1

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.

Answer 2

We have

$$f(x)=\sum_{k=1}^\infty \frac{2^kx^{2^k-1}}{1+x^{2^k}}$$

and by root test

$$\sqrt[k]{\frac{2^kx^{2^k-1}}{1+x^{2^k}}}=2\sqrt[k]{\frac{x^{2^k-1}}{1+x^{2^k}}}$$

and for $|x|<1$

$$2\sqrt[k]{\frac{x^{2^k-1}}{1+x^{2^k}}} \to 0$$

$|x|\ge1$

$$2\sqrt[k]{\frac{x^{2^k-1}}{1+x^{2^k}}} \to 2$$