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Math Help - [SOLVED] Clarification on harmonic series

  1. #1
    Senior Member Pinkk's Avatar
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    [SOLVED] Clarification on harmonic series

    I know that \sum_{n=1}^{\infty}\frac{1}{n} diverges, but does \sum_{n=k}^{\infty}\frac{1}{n} diverge for any k > 1? It's been a while since I've dealt with series.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Pinkk View Post
    I know that \sum_{n=1}^{\infty}\frac{1}{n} diverges, but does \sum_{n=k}^{\infty}\frac{1}{n} diverge for any k > 1? It's been a while since I've dealt with series.
    Yes, since \sum_{n=k}^{\infty}\frac{1}{n}={\color{red}\sum_{n  =1}^{\infty}\frac{1}{n}}-\sum_{n=1}^{k-1}\frac{1}{n} where we have the harmonic series appear again...
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  3. #3
    Senior Member Pinkk's Avatar
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    Ah okay, thanks. That jogs my memory a bit.
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  4. #4
    Math Engineering Student
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    \sum\limits_{n=1}^{\infty }{\frac{1}{n}}=1+\frac{1}{2}+\cdots +\sum\limits_{n\ge k}{\frac{1}{n}}, since the series of the LHS diverges, then obviously \sum_{n\ge k}\frac1n diverges.

    (ohh, i'm a little slow today!)
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