Does $\sum\limits_{n=1}^{\infty}\frac{x}{1+n^4x^2}$ converge uniformly? This is a very simple question and still I have a problem with that. I want to check whether this series converges   $\sum\limits_{n=1}^{\infty}\frac{x}{1+n^4x^2}$ where $x \in [0, \infty]$
This is an attempt to use the M-Weierstrass test for uniform convergence:
$\sum_{n=1}^{\infty}|\frac{x}{1+n^4x^2} |\leq \sum_{n=1}^{\infty}|\frac{x}{n^4x^2}| \leq \sum_{n=1}^{\infty}|\frac{1}{n^4x}|\leq \frac{1}{|x|} \sum_{n=1}^{\infty}|\frac{1}{n^4}|$
when $x$ is not 0, bus still fixed. For 0, It's easy to see that there's a uniform convergence. I feel that something is wrong with this use of this test. What is wrong? and What is the correct way showing the uniform convergence?

Since, for every $x$ in $\mathbb R$ and integer $n\geqslant1$,
$$
\left|\frac{x}{1+n^4x^2}\right|\leqslant\frac1{2n^2},
$$
the series is normally convergent on $\mathbb R$. In particular, it is uniformly convergent on $\mathbb R$.
