Another beautiful sum

We can extimate $\displaystyle \sum_{k=1} ^{10000} \frac 1{\sqrt k}$ using $\displaystyle \int_a^b x^{-\frac 12}\ dx$ . We note that:

$\begin{aligned} \int_1^{10000} x^{-\frac 12}\ dx < & \sum_{k=1}^{10000} \frac 1{\sqrt k} < \int_{0.5}^{10000.5} x^{-\frac 12}\ dx \\ 2x^\frac 12 \bigg|_1^{10000} < & \sum_{k=1}^{10000} \frac 1{\sqrt k} < 2x^\frac 12 \bigg|_{0.5}^{10000.5} \\ 198 < & \sum_{k=1}^{10000} \frac 1{\sqrt k} < 198.5907 \end{aligned}$

Therefore $\displaystyle \left \lfloor \sum_{k=1}^{10000} \frac 1{\sqrt k}\right \rfloor = \boxed{198}$ .