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Math Help - Harmonic numbers proof

  1. #1
    Senior Member Danneedshelp's Avatar
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    Apr 2009
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    Harmonic numbers proof

    Q: Prove H_{1}+H_{2}+...+H_{n}=(n+1)H_{n}-n.

    A: First of all, this is only true for n>1. So, I let n=2 for my base case. The equation held, so I moved on.

    Now, I assume H_{1}+H_{2}+...+H_{k}=(k+1)H_{k}-k for some natrual number k.

    Next, I have that

    H_{1}+H_{2}+...+H_{k}+H_{k+1}=((k+1)H_{k}-k)+H_{k+1} by our inductive hypothesis.

    I expanded everything on the RHS to get

    (k+1)(1+\frac{1}{2}+...+\frac{1}{k})-k+(1+\frac{1}{2}+...+\frac{1}{k}+\frac{1}{k+1})
    (k+1)+\frac{k+1}{2}+...+\frac{k+1}{k}-k+(1+\frac{1}{2}+...+\frac{1}{k}+\frac{1}{k+1})
    (k+2)+\frac{k+2}{2}+...+\frac{k+2}{k}-k+\frac{1}{K+1}
    (k+2)(1+\frac{1}{2}+...+\frac{1}{k})-k+\frac{1}{k+1}

    Now, I am not sure what to do. I think I may be taking the wrong approach to the problem. Any help would be great.

    Thanks
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  2. #2
    Member Traveller's Avatar
    Joined
    Sep 2010
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    (k+1)H_k - k + H_{k+1} = (k+1)(H_{k+1} - \frac{1}{k+1}) - k + H_{k+1}

    Can you complete the proof ?
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