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Math Help - inequality

  1. #1
    Junior Member
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    inequality

    Can anybody help me with the proof of this inequality please?

    sum_{k=1}^n(a_k).sum_{k=1}^n(1/a_k)<=n^2
    a_k>0

    thank you
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  2. #2
    MHF Contributor red_dog's Avatar
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    The correct inequality is this:

    \left(\sum_{k=1}^na_k\right)\left(\sum_{k=1}\frac{  1}{a_k}\right)\geq n^2

    Use the AM-GM inequality.

    \sum_{k=1}^na_k\geq n\sqrt[n]{a_1a_2\ldots a_n}

    \sum_{k=1}^n\frac{1}{a_k}\geq n\cdot\frac{1}{\sqrt[n]{a_1a_2\ldots a_n}}

    Now multiply the inequalities, member with member.
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