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Powerseries of a natural logarithm

  1. Oct 11, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that

    [tex]\frac{1}{x} \ln \frac{x+1}{x-1} = \sum_0^\infty \frac{2x^{2n}}{2n+1}[/tex].


    2. The attempt at a solution
    I tried to use the relation

    [tex]\frac{1}{1+x} = \frac{d}{dx}\ln (1+x)[/tex]

    and expand as a geometric series, but this did not lead anywhere since I then ended up with an alternating series.

    Anyone got any ideas of where to start?
     
  2. jcsd
  3. Oct 11, 2011 #2

    Hootenanny

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    HINT:

    [tex]\log\left(\frac{a}{b}\right) = \log a - \log b[/tex]
     
  4. Oct 11, 2011 #3
    Yes, so I could subtract the two resulting series... and I see now that it will actually lead trough. Ofcourse I knew about the property, but I did not see how it would get rid of the 'alternation'. However I do see that now because of the cancellation of terms.

    Thank you!
     
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