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Math Help - uniform convergence of series of functions

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by doxian View Post
    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
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  3. #3
    Senior Member Shanks's Avatar
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    the answer is nagetive.
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  4. #4
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    Re:

    Quote Originally Posted by dhammikai View Post
    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.
    Last edited by doxian; December 3rd 2009 at 02:09 AM.
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  5. #5
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    Quote Originally Posted by Shanks View Post
    the answer is nagetive.
    Thanks a lot. Can you please give me a counterexample?
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  6. #6
    Senior Member Shanks's Avatar
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    counterexample:
    f_n(x)=\frac{\frac{x}{n}}{1+(\frac{x}{n})^2}
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