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Need some series/ summation help

  1. Aug 11, 2009 #1
    [tex]\sum_{n=1}^{\infty}(-1)^{n}\frac{e^{-\frac{1}{nx}}}{n}[/tex]

    Where 0<x<oo.

    I'm looking for a closed form/ closed representation for this series [I was thinking something like a polylogarithm or dirichlet eta function combination might work].

    Any ideas or suggestions would be much appreciated.
     
  2. jcsd
  3. Aug 13, 2009 #2
    it does not converge...
     
  4. Aug 13, 2009 #3
    Sure it does, it differs from [tex]\sum_{n=1}^\infty \frac{(-1)^n}{n}[/tex] by an absolutely convergent series.
     
  5. Aug 13, 2009 #4
    If (-1)^n is being raised to [tex]\frac{e^{-\frac{1}{nx}}}{n}[/tex], i do not believe it converges. (it can also be simplified, the n's go away). please clarify what you mean.
     
  6. Aug 14, 2009 #5
    I understand the confulsion. If you read the TeX code included, you can see what was actually written. The term to be summed is [itex](-1)^n[/itex] times a fraction:
    [tex]\sum_{n=1}^{\infty}\;(-1)^{n}\left(\frac{e^{-\frac{1}{nx}}}{n}\right)[/tex]
     
  7. Aug 16, 2009 #6
    Sorry about the confusion. I should have included the brackets as you demonstrated.
     
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