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Homework Help: Converges uniformly

  1. Sep 10, 2011 #1
    1. The problem statement, all variables and given/known data
    [itex]I = [0, \frac{\Pi}{2}][/itex] is an interval, and [itex]\lbrace f_n(x)\rbrace_{n=1}^{\infty}[/itex] is sequence of continuous function. [itex]\sum_{n=1}^{\infty}f_n(x)[/itex] converges uniformly on the interval[itex]I[/itex] .
    Show that holds

    [itex] \int_{0}^{\frac{\Pi}{2}} \sum_{n=1}^{\infty} f_n(x)dx = \sum_{n=1}^{\infty} \int_{0}^{\frac{\Pi}{2}} f_n(x)dx [/itex]


    2. Relevant equations



    3. The attempt at a solution
    I`m not familiar that how we show this with using uniformly converge????
    pls help, I will be appreciate...
     
  2. jcsd
  3. Sep 10, 2011 #2
    can we say that?

    If [itex]\sum_{n=1}^{\infty}f_n(x)[/itex] converges uniformly on [itex]I=[0, \frac{\Pi}{2}][/itex],
    then we can write,

    [itex]\int_0^{\frac{\Pi}{2}} \sum_{n=1}^{\infty}f_n(x)dx = \int_0^{\frac{\Pi}{2}} \lim_{\frac{\Pi}{2} \rightarrow \infty} S_n(x)dx = \lim_{\frac{\Pi}{2} \rightarrow \infty} \int_0^{\frac{\Pi}{2}} S_n(x)dx [/itex]
    [itex]~~~=\sum_{n=1}^{\infty} \int_0^{\frac{\Pi}{2}} f_n(x)dx [/itex]

    ??????
     
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