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Find constants s.t. the following expression holds for all n.

  1. Oct 4, 2009 #1
    1. The problem statement, all variables and given/known data

    Find constants [tex]c_1,c_2[/tex] (independent of n) such that the following holds for all [tex]n\in \mathbb{N}[/tex]:

    [tex]\left| \sum^{2n}_{k=n+1} \frac{1}{k} - \log 2 - \frac{c_1}{n} \right| \le \frac{c_2}{n^2}.[/tex]

    2. Relevant equations

    [tex]\log(2) = \sum^{\infty}_{k=1} (-1)^{k+1}\left( \frac{1}{k} \right). [/tex]

    3. The attempt at a solution

    Well, the obvious method would be to just input the series for [tex]\log(2)[/tex] and try to find something from that. I tried but found virtually nothing that I could work with. Can anyone give me any hints as to how to proceed?

    Edit: Okay, I think I have just confirmed that

    [tex] \sum^{2n}_{k=n+1} \frac{1}{k} - \log 2 = -\sum^{\infty}_{k=2n+1} (-1)^{k+1} \left( \frac{1}{k} \right). [/tex]
     
    Last edited: Oct 4, 2009
  2. jcsd
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