Uniform Convergence for $\sum\limits_{n=1}^{\infty}{\frac{x^{n}}{100+x^{2n}}}$

Question

Prove that $$ \sum_{n=1}^\infty f_n(x) = \sum_{n=1}^{\infty}{\frac{x^{ n}}{100+x^{2n}}}$$ converges uniformly for $x \in [2,\infty)$?

I tried to prove it by applying Weierstrass M-test, where the $M_n$ that I found to be larger or equal to the sum above would be $$\sum_{n=1}^{\infty} {\frac{1}{nx}}$$ but I am not sure if it is an appropriate sum to use.

Answer 1

$0 <\frac {x^{n}} {100+x^{2n}} \leq \frac {x^{n}} {x^{2n}} =\frac 1 {x^{n}} \leq \frac 1 {2^{n}}$ and $\sum \frac 1 {2^{n}}<\infty$.