uniform convergence of series of functions

Hi,

I am really stuck with this. Any help will be really, really welcome.

Suppose I can show a series of functions (for example $\displaystyle \sum_{n=1}^{\infty}\frac{nx^2}{n^3+x^3}$) is uniformly convergent on [0,B] for all B>0. Does this mean that the series is uniformly convergent of [0, \infty)??

Generalizing, suppose a sequence of functions is uniformly convergent for all compact subsets, is it convergent on the whole domain?

Thanks a lot in advance