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Math Help - Please help me

  1. #1
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    Feb 2010
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    6

    Please help me

    Question in the Attach Files, thanks
    Attached Files Attached Files
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  2. #2
    Senior Member
    Joined
    Feb 2008
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    Quote Originally Posted by ha11 View Post
    Show that \sum_{k=1}^n\frac{1}{k}=O(\log n)

    and \int_1^{\infty}\frac{\lfloor t\rfloor}{t^2}dt=\sum_{1\leq r\leq t\leq n}\int_1^{\infty}\frac{dt}{t^2}=\sum_{r=1}^n\left(  \frac{1}{r}-\frac{1}{n}\right).
    It may help others to see this without the attachment.

    Now, as for solutions.... regarding the second, where does n come from? I can only show that

    \int_1^{\infty}\frac{\lfloor t\rfloor}{t^2}dt=\sum_{i=1}^{\infty}\int_i^{i+1}\f  rac{i}{t^2}dt

    =\sum_{i=1}^{\infty}i\int_i^{i+1}t^{-2}dt

    =\sum_{i=1}^{\infty}i\left[-t^{-1}\right]_i^{i+1}

    =\sum_{i=1}^{\infty}i\left(i^{-1}-(i+1)^{-1}\right)

    =\sum_{i=1}^{\infty}\frac{1}{i+1}

    =\sum_{i=1}^{\infty}\frac{1}{i}-1.
    Last edited by hatsoff; February 1st 2010 at 04:13 PM.
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  3. #3
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    thanks my friend.
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