Sum of square roots........... $$
\mbox{If}\quad S =
1 + \,\sqrt{\,\frac{1}{2}\,}\, + \,\sqrt{\,\frac{1}{3}\,}\, +
\,\sqrt{\,\frac{1}{4}\,}\, +
\,\sqrt{\,\frac{1}{5}\,}\, + \cdots + \,\sqrt{\,\frac{1}{100}\,}\,\,,
$$
then what is the value of $\left\lfloor\,S\,\right\rfloor$ ?.
Here $\left\lfloor\,S\,\right\rfloor$ is the greatest integer function which is less than or equal to $S$.

The Euler-Maclaurin Summation Formula yields
$$\begin{align}
\sum_{k=1}^{100}k^{-1/2}&=1 +\int_1^{100} x^{-1/2}\,dx+\frac12\left(\frac{1}{\sqrt{100}}-1\right)+\frac1{24} \left(1-\frac{1}{(100)^{3/2}}\right)+R_2 \\\\
&=18.591625+R_2
\end{align}$$
where 
$$\begin{align}
\left|R_2\right|&\le \frac{2\zeta(2)}{(2\pi)^2}\int_1^{100} \left|\frac{d^2 x^{-1/2}}{dx^2}\right|\,dx\\\\
&= \frac{2\zeta(2)}{(2\pi)^2}\,\frac12 \left(1-\frac{1}{(100)^{3/2}}\right)\\\\
&=0.041625
\end{align}$$
Therefore, the integer part of the sum of interest is indeed $18$.
