According to W.A., the series expansion at $n=\infty$ for the generalized harmonic series $H_n^{(-1/2)}$ is $$H_n^{(-1/2)}=\frac{2n^{3/2}}3+\frac{\sqrt n}2+\zeta\left(-\frac{1}2\right)+\frac{1}{24\sqrt{n}}+O(1/n).$$
How would one go about proving this? The only thing I know about this function is that it is defined by $H_n^{(-1/2)}=\sum_{k=1}^n\sqrt k$.