convergence of $\sum \frac {nx^{n-1}} {(1+x^n)^2}$

Question

I want to examine the convergence of this function series $\sum \frac {nx^{n-1}} {(1+x^n)^2}$ for $x \in [2, +\infty)$. I showed pointwise convergence but I'm struggling with uniform convergence.

I tried to apply the Weierstrass M-test but it didn't work for me. I also tried to demonstrate that the series does not converge uniformly but that didn't work either. Does anybody have any hints?

Answer 1

Hint: $$\frac {nx^{n-1}} {(1+x^n)^2} \le \frac {nx^{n-1}}{x^{2n}} = \frac {n} {x^{n+1}} \le \frac {n} {2^{n+1}}.$$