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Math Help - proof of harmonic series

  1. #1
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    proof of harmonic series

    Find an proof for the harmonic series not using the integral test with the following hint.

    \sum_{k = 1}^{\infty} = 1 + (\frac{1}{2} + \frac{1}{3}) + (\frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7}) + ...

    observe that

    \frac{1}{2} + \frac{1}{3} > \frac{1}{4} + \frac{1}{4} = \frac{1}{2}

    \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} > \frac{1}{8}+ \frac{1}{8}+ \frac{1}{8}+ \frac{1}{8} =  \frac{1}{2}

    It looks like we are comparing grouping of the harmonic series to another series. Each grouping is always grater than 1/2 so the harmonic series is greater than 1 + \sum_{k = 1}^{\infty} \frac{1}{2} which diverges.

    1 + \sum_{k = 1}^{\infty} \frac{1}{2} = 1 + \frac{1}{2} +  \frac{1}{2} +  \frac{1}{2} + ...

    Therefor the harmonic series diverges.
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  2. #2
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    Quote Originally Posted by diroga View Post
    Find an proof for the harmonic series not using the integral test with the following hint.

    \sum_{k = 1}^{\infty} = 1 + (\frac{1}{2} + \frac{1}{3}) + (\frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7}) + ...

    observe that

    \frac{1}{2} + \frac{1}{3} > \frac{1}{4} + \frac{1}{4} = \frac{1}{2}

    \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} > \frac{1}{8}+ \frac{1}{8}+ \frac{1}{8}+ \frac{1}{8} =  \frac{1}{2}

    It looks like we are comparing grouping of the harmonic series to another series. Each grouping is always grater than 1/2 so the harmonic series is greater than 1 + \sum_{k = 1}^{\infty} \frac{1}{2} which diverges.

    1 + \sum_{k = 1}^{\infty} \frac{1}{2} = 1 + \frac{1}{2} +  \frac{1}{2} +  \frac{1}{2} + ...

    Therefor the harmonic series diverges.
    That is correct.
    You may be a bit more formal. But the idea is correct.
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  3. #3
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    is using <br /> <br />
1 + \sum_{k = 1}^{\infty} \frac{1}{2} = 1 + \frac{1}{2} +  \frac{1}{2} +  \frac{1}{2} + ...<br />
an acceptable comparison
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