Question

$\underset{n\to \infty }{\lim }\left(\frac{1}{\sqrt{n}\sqrt{n+1}}+\frac{1}{\sqrt{n}\sqrt{n+2}}+\dots +\frac{1}{\sqrt{n}\sqrt{4n}}\right)$ is equal to

Answer 1

Answer: 2

$\underset{n\to \infty }{\lim }\sum _{r=1}^{r=3n}\frac{1}{\sqrt{n}\sqrt{n+r}}$
$\underset{n\to \infty }{\lim }\sum _{r=1}^{r=3n}\frac{\frac{1}{n}}{\sqrt{1+\frac{r}{n}}}=\underset{0}{\overset{3}{\int }}\frac{dx}{\sqrt{1+x}}=2$