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Math Help - Convergence proof...

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
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    Convergence proof...

    Check the convergence of:

    \displaystyle \sum^{\infty}_{n=1} \int^{\frac{sin(n)}{n}}_0 \frac{sin(x)}{x}dx


    Thank you!
    Last edited by CaptainBlack; July 1st 2010 at 05:21 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Check the convergence of:

    \displaystyle \sum^{\infty}_{n=1} \int^{\frac{sin(n)}{n}}_0 \frac{sin(x)}{x}dx


    Thank you!
    As $$ n becomes large the intervals of integration contract about 0, where the integrand is close to 1. So we need only consider the convergence of:

    \displaystyle \sum^{\infty}_{n=1} \int^{\frac{sin(n)}{n}}_0 dx

    When \sin(n)<0 the corresponding term in the sum is |\sin(n)|/n and when \sin(n)>0 the corresponding term is \sin(n)/n

    So we need only consider the convergence of:

    \displaystyle \sum^{\infty}_{n=1}\frac{|sin(n)|}{n}

    which I'm pretty sure diverges, but will leave the proof of that to the reader

    CB
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