Sum of Square roots formula.

The definition of harmonic numbers is $$H_p^{(-a)}=\sum_{i=1}^p i^a$$ When $a$ is not a positive integer, there is no closed form but, as Yves Daoust commented, there are quite nice expansions.

For example, if $n=\frac 12$ as in the post, you have $$H_p^{\left(-\frac{1}{2}\right)}=\frac{2 p^{3/2}}{3}+\frac{\sqrt{p}}{2}+\zeta \left(-\frac{1}{2}\right)+\frac{1}{24\sqrt p}+O\left(\left(\frac{1 }{p}\right)^{5/2}\right)$$ where $\zeta \left(-\frac{1}{2}\right)\approx -0.2078862250$.

For example, for $p=10$, the exact value is $\approx 22.46827819$ while the above approximation gives $\approx 22.46827983$. By itself, the first term already gives $21.0819$; the sum of first and second term gives $\approx 22.6629$. For $p=100$, the approximation leads to $12$ exact significant figures.

There are similar expansions for any value of the exponent $a$