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 $S$.

Hint:
$$S=\sum_{k=1}^{100}\frac{1}{\sqrt{k}}$$
$$\frac{1}{\sqrt{k+1}+\sqrt{k}}\lt \frac{1}{2\sqrt{k}}\lt \frac{1}{\sqrt{k+\frac{1}{2}}+\sqrt{k-\frac{1}{2}}}$$
$$\implies \sqrt{k+1} -\sqrt{k} \lt \frac{1}{2\sqrt{k}}\lt \sqrt{k+\frac12}-\sqrt{k-\frac{1}{2}}$$
