Question 7
In this paper, let $[x]$ denote the greatest integer not exceeding x. For examples, $[5]=5$, $[2.8]=2$, and $[-2.3]=-3$.
Find $[S/2]$, where $$S=\frac 1{\sqrt 1}+\frac 1{\sqrt 2}+\cdots+\frac 1{\sqrt{100}}.$$
(Hint: Note that $\frac 2{\sqrt k+\sqrt{k+1}}\lt \frac 1{\sqrt k}\lt \frac 2{\sqrt k+\sqrt{k-1}}$.)
Find $[S/2]$, where $$S=\frac 1{\sqrt 1}+\frac 1{\sqrt 2}+\cdots+\frac 1{\sqrt{100}}.$$
(Hint: Note that $\frac 2{\sqrt k+\sqrt{k+1}}\lt \frac 1{\sqrt k}\lt \frac 2{\sqrt k+\sqrt{k-1}}$.)