Question

$\displaystyle\lim _{n \rightarrow \infty}\left(\frac{1}{\sqrt{n} \sqrt{n+1}}+\frac{1}{\sqrt{n} \sqrt{n+2}}+\ldots+\frac{1}{\sqrt{n} \sqrt{4 n}}\right)$ is equal to

Answer 1

$\displaystyle\lim _{n \rightarrow \infty} \displaystyle\sum_{r=1}^{r=3 n} \frac{1}{\sqrt{n} \sqrt{n+r}}$
$\displaystyle\lim _{n \rightarrow \infty} \displaystyle\sum_{r=1}^{r=3 n} \frac{\frac{1}{n}}{\sqrt{1+\frac{r}{n}}} = \int\limits_0^3 \frac{d x}{\sqrt{1+x}}=2$