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?

• I think that's correct for $|x|\geq1$ – fiorerb Feb 10 '12 at 15:11

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 $\mathbb R$.
• Should I need to show that just $a_n$ is a monotic function for $\sum_{n=1}^{\infty} a_n$? – Jozef Feb 10 '12 at 15:39
• $[0,\infty)$ is not compact. – Najib Idrissi Feb 10 '12 at 16:23