# uniform convergence of series of functions

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• December 3rd 2009, 12:08 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?

Thanks a lot in advance
• December 3rd 2009, 12:56 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?

Thanks a lot in advance

Use ratio test
• December 3rd 2009, 01:48 AM
Shanks
the answer is nagetive.
• December 3rd 2009, 01:56 AM
doxian
Re:
Quote:

Originally Posted by dhammikai
Use ratio test

Thanks for your reply. But I wasn't asking for any particular series of functions, I was asking the general question:
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, 02:02 AM
doxian
Quote:

Originally Posted by Shanks
the answer is nagetive.

Thanks a lot. Can you please give me a counterexample?
• December 3rd 2009, 03:59 AM
Shanks
counterexample：
$f_n(x)=\frac{\frac{x}{n}}{1+(\frac{x}{n})^2}$