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Limit of an integral

  1. May 19, 2009 #1
    1. The problem statement, all variables and given/known data
    [tex]lim n \rightarrow\inf \int sin(pi*x^{n})dx[/tex]
    ...integral is from x=0 to 1/2.


    2. Relevant equations



    3. The attempt at a solution
    Lebesgue's Dominated Convergence Theorem says that I can move the limit inside, but only if fn converges pointwise to a limit f, which I don't believe it does. Even so, there is no limit as n approaches infinity of fn.
    I also tried u substitution, setting u = pi*x^n, but that didn't get me anywhere.

    Thanks in advance
     
  2. jcsd
  3. May 19, 2009 #2

    Dick

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    Doesn't x^n converge to zero for x in [0,1/2]? Or am I confused?
     
  4. May 19, 2009 #3
    It does, but in order to move the limit inside and use Lebesgue's, doesn't sin(pi*x^n) have to converge to a limit over the entire domain, not just [0,1/2]?
     
  5. May 19, 2009 #4

    Dick

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    Not as far as I know. You are only integrating over [0,1/2]. Why do you have to worry about values outside of that range? Just call the domain [0,1/2].
     
  6. May 19, 2009 #5
    I guess I was over thinking it. Thanks for your help.
     
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