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Math Help - Proof of summation 1/i^2 <= 2 - 1/n

  1. #1
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    Proof of summation 1/i^2 <= 2 - 1/n

    I have put up an attachment .doc file with my work. I would like to know if my proof is correct.

    Thanks
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  2. #2
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    there is a short way to prove this by using some of calculus.

    are you allowed to use some of calculus?
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  3. #3
    Super Member PaulRS's Avatar
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    Quote Originally Posted by Bruins1 View Post
    I have put up an attachment .doc file with my work. I would like to know if my proof is correct.

    Thanks
    It's not correct. You say that \sum_{k=1}^{n}{\tfrac{1}{k^2}}=\tfrac{\pi^2}{6} when that is not so, actually: \sum_{k=1}^{+\infty}{\tfrac{1}{k^2}}=\tfrac{\pi^2}  {6}, besides you use approximations and not the real value of the sum.

    Note that: \tfrac{1}{k^2}<\tfrac{1}{k\cdot (k-1)}=\tfrac{1}{k-1}-\tfrac{1}{k} for k>1

    What do you see if you sum on both sides from k=2 to k=n ? - add afterwards any missing terms-
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