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Function defined as series

  1. Dec 7, 2013 #1
    Hi, does anyone know if this function:

    [tex] f(x) = \sum_{k=1}^\infty \frac{(-1)^n}{x^{2k}} [/tex]

    is representable as an elementary or already defined special function? Thanks
     
  2. jcsd
  3. Dec 7, 2013 #2
    This function can be broken up as:

    [tex] f(x) = \sum_{k=1}^\infty x^{4k} - \sum_{n=1}^\infty x^{4n-2} [/tex]

    Any ideas?
     
  4. Dec 7, 2013 #3
    I think I've got it: the above expression is equal to

    [tex] \frac{1}{x^4 - 1} - \frac{ x^2} {x^4 - 1} = - \frac{1}{1 + x^2} [/tex]

    does that look ok?
     
  5. Dec 7, 2013 #4

    mathman

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    Gold Member

    Answer is correct. It is a geometric series, ratio = -1/x2.
     
  6. Dec 7, 2013 #5
    Oh, right. I'm an idiot for not seeing that to begin with
     
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