I'm wondering whether this sum has a closed form expression:
$$\frac{1}{k+1}\sum^{r_0+k}_{r=r_0}\frac{{(p - r)}^{0.75}}{r^2}$$
I can do this: Let $z = p - r$ then $r = p - z$
$$\frac{1}{k+1}\sum^{p - r_0 - k}_{z=p - r_0}\frac{{z}^{0.75}}{(p - z)^2}$$
$$\frac{1}{k+1}\sum^{p - r_0 - k}_{z=p - r_0}\frac{{z}^{0.75}}{p^2 - 2zp + z^2}$$
But then it's unclear where to go from here. Any ideas? I have no idea if there even is a solution for this. This does not look like the series I have studied in calculus. Does this form of summation have a name? Is it known to have a closed form?