# Sum of Square roots formula.

I would like to know if there is formula to calculate sum of series of square roots $\sqrt{1} + \sqrt{2}+\dotsb+ \sqrt{n}$ like the one for the series $1 + 2 +\ldots+ n = \frac{n(n+1)}{2}$.

The definition of harmonic numbers is 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 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 $a$