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). 

There is an asymptotic Euler-Maclaurin expansion for generalized harmonic numbers:
$$\zeta(s)\sim H_{N,s}+N^{-s}\sum_{k\,\ge-1}\frac{B_{k+1}}{(k+1)!}\frac{s^{(k)}}{N^k} \quad \textrm{as }~N\to\infty$$
The $B_{k+1}$s are the Bernoulli numbers with $B_1:=+\frac{1}{2}$ and the $s^{(k)}=s(s+1)\cdots(s+(k-1))$ are rising factorials. One may of course isolate the $H_{N,s}$ term for an asymptotic with it instead.
Note $H_{N,s}:=\displaystyle\sum_{k=1}^N k^{-s}$ and $s\in\Bbb C\setminus\{1\}$. Source: e.g. the first equation here.
