I think I'm able to show show if $f_n(x)$ defined on $R$ by $f_n(x)=\frac{1}{n+n^2x}$ is simply but not uniformly converging.
$\lim_\limits{n\rightarrow+\infty}\frac{1}{n+n^2x}=\frac{1}{n^2}$ (but I'm not sure why... only intuition...)
Yet, $\sum\frac{1}{n^2}$ is converging as a geometrical series. Thus $\sum f_n(x)$ converges simply.
Then how to show that it is uniformly converging?