# uniform convergence of series of functions

• December 3rd 2009, 01:08 AM
doxian
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 $\sum_{n=1}^{\infty}\frac{nx^2}{n^3+x^3}$) 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?

• December 3rd 2009, 01:56 AM
dhammikai
Quote:

Originally Posted by doxian
Hi,

I am really stuck with this. Any help will be really, really welcome.
Suppose I can show a series of functions (for example $\sum_{n=1}^{\infty}\frac{nx^2}{n^3+x^3}$) 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?

Use ratio test
• December 3rd 2009, 02:48 AM
Shanks
• December 3rd 2009, 02:56 AM
doxian
Re:
Quote:

Originally Posted by dhammikai
Use ratio test

Is it true that : If a series of function converges uniformly on [0, B] for all B>0 then the series is uniformly convergent on [0, \infty)

Sorry for not being clear.

Add: Unfortunately, the Ratio test doesn't work for the example I gave. The limit goes to 1.
• December 3rd 2009, 03:02 AM
doxian
Quote:

Originally Posted by Shanks
$f_n(x)=\frac{\frac{x}{n}}{1+(\frac{x}{n})^2}$