Summation of roots In this post, Calculate summation of square roots we are shown how to sum square roots. My question is, can we get similarly simple expressions if instead of square roots we choose some other exponent. More specifically, is there a simple approximating expression for
$\sum_{i=1}^N i^{2/3}$

Well writing $\;\displaystyle\sum_{k=1}^N k^a=H_N^{\left(-a\right)}\;$ is just a rewriting using generalized harmonic numbers.
Writing this as a partial zeta sum or truncated zeta function $\zeta_N(-a)$ is exactly the same thing!
You get too $\;\displaystyle\sum_{k=1}^N k^a =\zeta(-a) - \zeta(-a, N+1)\;$ using the Hurwitz zeta function.
For $a$ integer you get Faulhaber's formula and for real or even complex values $\,a\neq -1\,$ excellent approximations are provided by the Euler-Maclaurin formula (with $s=-a$) :
$$\sum_{k=1}^N k^a \sim\zeta(-a)+\frac {N^{a+1}}{a+1}+\frac {N^a}2-\sum_{i\ge 1}\frac{B_{2i}}{(2i)!}\frac{(s+2i-2)!\;N^{a-2i+1}}{(s-1)!}$$
(note that this is an asymptotic expansion and that the sum at the right has to remain finite with $B_j$ a Bernoulli numbers)
