I've these two series, and I would like a closed form:
$ \sum_{k=-\infty}^{\infty} \frac{x+kx_0-h}{|x+kx_0|^3}$
$ 3\sum_{k=-\infty}^{\infty} \frac{(x+kx_0-h)(x+kx_0)^2}{|x+kx_0|^5} $
Mathematica gives me these closed forms ($A=x_0$, $B=h$):
Sum[(x + k*A - B)/((abs (x + k*A))^3), {k, -Infinity, Infinity}]
= (1/(2 A^3 abs^3))(2 A PolyGamma[1, x/A] + 2 A PolyGamma[1, 1 - x/A] + B PolyGamma[2, x/A] - B PolyGamma[2, 1 - x/A])
Sum[((x + k*A - B)*(x + k*A)^2)/(abs (x + k*A)^5), {k, -Infinity, Infinity }]
= (1/(2 A^3 abs))(2 A PolyGamma[1, x/A] + 2 A PolyGamma[1, 1 - x/A] + B PolyGamma[2, x/A] - B PolyGamma[2, 1 - x/A])
Now, first of all, when it writes $abs^3$ this should mean $|x|^3$ or $|x-kx_0|^3$? And at last, does this make sense? Is there an analytical way to reach this (or a better) conclusion?
Thanks!