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Math Help - Prove that harmonic series diverge

  1. #1
    Senior Member Twig's Avatar
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    Prove that harmonic series diverge

    Hi

    Problem: Consider  \sum_{k=1}^{\infty} \frac{1}{k} . With S_{n}=\sum_{k=1}^{n} \frac{1}{k} , prove that S_{2^{n}}\geq 1+\frac{n}{2} by writing S_{2^{n}}=1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}  {4}\right) + ... + \left(\frac{1}{2^{n-1}+1}+...+\frac{1}{2^{n}}\right)

    Need some guidelines.

    Thx!
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  2. #2
    Flow Master
    mr fantastic's Avatar
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    Quote Originally Posted by Twig View Post
    Hi

    Problem: Consider  \sum_{k=1}^{\infty} \frac{1}{k} . With S_{n}=\sum_{k=1}^{n} \frac{1}{k} , prove that S_{2^{n}}\geq 1+\frac{n}{2} by writing S_{2^{n}}=1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}  {4}\right) + ... + \left(\frac{1}{2^{n-1}+1}+...+\frac{1}{2^{n}}\right)

    Need some guidelines.

    Thx!
    Read this: http://en.wikipedia.org/wiki/Harmoni...s_(mathematics)
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