Asymptotic of $ \sum_{r=1}^{k} \frac1{r^{3/2}}$ - Generalized Harmonic Number What is the asymptotic approximation of the following generalized Harmonic number as $k \to \infty$ ? 

$$H(1.5,k) = \sum_{r=1}^{r=k} \frac{1}{r^{1.5}}$$

(there is a similar question posted on MSE but there is a wrong comment bellow which refers the reader to wolfram alpha, the comment is wrong since the notation used on wolfram alpha is different from what the questioner used). 

Hint. You may express you sum in terms of the Hurwitz zeta function:
$$
\sum_1^k\frac{1}{r^{3/2}}=\zeta(3/2)-\zeta(3/2,k+1)
$$ and the asymptotic behavior, as $k \to \infty$, is given by

$$
\sum_1^k\frac{1}{r^{3/2}}= \zeta(3/2)-\frac{2}{\sqrt{k+1}}-\frac12 \frac1{(k+1)^{3/2}}+\mathcal{O}\left( \frac1{k^{5/2}}\right)
$$ 

(see the link above). You would obtain the same result with the Euler-Maclaurin formula.
