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Does this infinite series make sense ?

  1. Apr 8, 2008 #1
    given the series

    [tex] g(x)= \sum_{n=0}^{\infty}\frac{a_{n}}{\sqrt {x-n}} [/tex]

    where the coefficients a_n are real numbers my question is does the above makes sense ? i mean since we are summing over all positive integers , no matter how big we choose 'x' there will be a factor so x-n n=0,1,2,3,4,.................. (x-n) <0 then g(x) is complex no matter what real x we put.

    If we approximate the series by an integral [tex] \int_{0}^{\infty}dr a(r) (x-r)^{-1/2} [/tex] we get a similar result, i do not know if this is paradoxical and i should have taken the absolute value |x| inside the square root.
  2. jcsd
  3. Apr 8, 2008 #2


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    Science Advisor

    It makes sense mathematically. You end up with a complex function. Convergence may be another matter - depending on the behavior of an.
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