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Math Help - [SOLVED] Prove 1 + 1/2 + 1/3 + ... + 1/n > 2n/n+1

  1. #1
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    [SOLVED] Prove 1 + 1/2 + 1/3 + ... + 1/n > 2n/n+1

    Prove that:

    1 + 1/2 + 1/3 + ... + 1/n > 2n/n+1

    where n > 2
    Last edited by mr fantastic; May 11th 2010 at 03:52 AM. Reason: Re-titled.
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  2. #2
    MHF Contributor red_dog's Avatar
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    Prove by induction.
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  3. #3
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    Proof by Induction

    Quote Originally Posted by the undertaker View Post
    Prove that:

    1 + 1/2 + 1/3 + ... + 1/n > 2n/n+1

    where n > 2
    This will be a proof by induction

    For the base case let n=2. 1+\frac{1}{2}>\frac{2(2)}{2+1}

    So \frac{3}{2}>\frac{4}{3} which is true so the base case holds

    Now we'll assume the statement is true for n and consider it for n+1

    \frac{1}{n+1}+\sum_{k=1}^n \frac{1}{k}>\frac{1}{n+1}+\frac{2n}{n+1}=\frac{2n+  1}{n+1}

    \frac{2(n+1)}{(n+1)+1}=\frac{2n+2}{n+2}

    Is \frac{2n+1}{n+1}>\frac{2n+2}{n+2}

    (2n+1)(n+2)=(2n^2+5n+2)
    (2n+2)(n+1)=(2n^2+4n+2) (this is an analysis via cross multiplication)

    And since 5n>4n for all n, \frac{2n+1}{n+1}>\frac{2n+2}{n+2} so the statement is proven by induction
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