Infinite Sum Question

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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.

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.
 

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