Question

Evalute $\underset{n\to \infty }{lim}\left(\frac{1}{\sqrt{n}\sqrt{n+1}}+\frac{1}{\sqrt{n}\sqrt{n+2}}+.......\frac{1}{\sqrt{n}\sqrt{n+n}}\right)$

Answer 1

$\underset{n\to \infty }{lim}(\frac{1}{\sqrt{n}\sqrt{n+1}}+\frac{1}{\sqrt{n}\sqrt{n+2}}+\cdots +\frac{1}{\sqrt{n}\sqrt{n+n}})=\underset{n\to \infty }{lim}\frac{1}{n}\sum _{r=1}^n\frac{1}{\sqrt{1+\frac{r}{n}}}={\int }_0^1(1+x{)}^{-1/2}=2[\sqrt{1+x}{]}_0^1=2\sqrt{2}-2$