Question

$\underset{n\to \infty }{\lim }$$\frac{3}{n}\left\{1+\sqrt{\frac{n}{n+3}}+\sqrt{\frac{n}{n+6}}+\sqrt{\frac{n}{n+9}}+....+\sqrt{\frac{n}{n+3\left(n-1\right)}}\right\}$

• A. does not exit
• B. is 1
• C. is 2
• D. is 3

Answer 1

Answer: (C)

$\underset{n\to \infty }{\lim }$$\frac{3}{n}\sum _{r=0}^{n-1}\sqrt{\frac{1}{1+3\left(\frac{r}{n}\right)}}$
$=3\underset{0}{\overset{1}{\int }}\frac{dx}{\sqrt{1+3x}}=2{\left[\sqrt{1+3x}\right]}_0^1=2$