Thread: 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 ) 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

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 ) 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

the answer is nagetive.

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.

Originally Posted by Shanks

the answer is nagetive.

Thanks a lot. Can you please give me a counterexample?

counterexample：