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Sum of a simple series

  1. Jun 16, 2012 #1
    1. The problem statement, all variables and given/known data

    sum this series:

    S = [itex] \sum_{-\infty}^\infty \frac{1}{|x-kx_0|} [/itex]


    2. Relevant equations



    3. The attempt at a solution

    [itex]S = \sum_{|x-kx_0|<0} \frac{1}{kx_0-x}+\sum_{|x-kx_0|>0}\frac{1}{x-kx_0} [/itex]

    but I don't know how to evaluate these two sums :(
     
  2. jcsd
  3. Jun 16, 2012 #2

    D H

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    Both of those series need to converge. Does either one do so?
     
  4. Jun 16, 2012 #3
    [itex] \sum_{k<A} \frac{1}{A-k} + \sum_{k>A} \frac{1}{k-A} [/itex]

    It looks like no, because they seems to behave as [itex] \sum \frac{1}{k} [/itex] (?)
     
  5. Jun 16, 2012 #4

    D H

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    Very good. You have to demonstrate this conclusively, but you are on the right track.
     
  6. Jun 16, 2012 #5
    Voilà, I think this is gonna work, so does the series converge? view attachement
     

    Attached Files:

  7. Jun 17, 2012 #6
    Based on that work, what would happen if n2x02 = x2??

    Could the series converge, then?
     
    Last edited: Jun 17, 2012
  8. Jun 17, 2012 #7
    According to this simulation with matlab, x=1, x_0=2, the series converges, but differently than that written in the last calculation (previous attachement) In blue the 1/x + 2x sum ....
     

    Attached Files:

  9. Jun 17, 2012 #8

    Ray Vickson

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    I don't think the series converges in the usual sense. Llet
    [tex]S(M,N) = \sum_{k=-M}^N \frac{1}{|x-k x_0|}.[/tex]
    The series converges if [itex] \lim_{M,N \rightarrow \infty} S(M,N)[/itex] exists and is finite. Note, however, that the M and N limits are separate, not coupled. What you have shown is that [itex] \lim_{N \rightarrow \infty} S(N,N)[/itex] exists and is finite, but that does not mean that S(M,N) converges.

    RGV
     
  10. Jun 17, 2012 #9
    Yes, actually I'm looking for an expression explicit in x of the series, so, whether if the series has limit or not, it can still be written like - for instance - something like [itex]1/x + log(x-x_0)[/itex] ?
     
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