<s>[INST] Uniform Convergence of $\frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+\frac{8x^7}{1+x^8}+...$ Show that following series is uniformly convergent in $(-1,1)$.
$$\frac{2x}{1+x^2}+\frac{4x^3}{1+x^4}+\frac{8x^7}{1+x^8}+...$$
I had previously solved this problem using Weierstrass M test by taking 
$$|f_n| = |\frac{2^n.x^{2^n-1}}{1+x^{2^n}}| \leq 2^n.k^{2^n-1}, k\in(x,1)$$
Thus, $\frac{M_n}{M_{n+1}}\to \infty$ hence the series is uniformly convergent. But now I think this is incorrect as $k$ shouldn't be dependent on $x$. How should I proceed then?
 [/INST] The series $\sum \frac{2^n.x^{2^n-1}}{1+x^{2^n}}$ is not uniformly convergent on $(-1,1)$. If it was $v_n = \sup\limits_{x \in (-1,1)} \left\vert \frac{2^n.x^{2^n-1}}{1+x^{2^n}} \right\vert$ would be such that $v_n \to 0$.
But 
$$\left\vert \frac{2^n.x^{2^n-1}}{1+x^{2^n}} \right\vert \ge 2^{n-1} \vert x \vert^{2^n-1}$$ and the right hand side of the inequality is unbounded in $(-1,1)$.
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