1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Infinite Sum Question

  1. Jan 28, 2009 #1
    1. I want to find a close expression for the following infinite sum

    [tex]\sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n^2+x_1 y}}-\frac{1}{\sqrt{n^2+x_2 y}}\right)[/tex]

    Both [tex]x_i[/tex] and [tex]y[/tex] are greater than 0.

    3. The attempt at a solution
    I haven't got much really. I realize each sum independently is divergent, but together they would be convergent. I just don't know how to extract the convergent part into a closed expression. I try doing a power expansion for y around 0 and reach

    [tex]\sum_{n=0}^{\infty}-\frac{\sqrt{\pi}Zeta(1+2n)(x_1^n-x_2^n)y^n}{\Gamma(\frac{1}{2}-n)\Gamma(1+n)n!} [/tex]

    but this is not much of an improvement over the original expression. Any help will be appreciated.
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted