Question

$\lim\limits_{n \to \infty}$ \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 exist
• B. is 1
• C. is 2
• D. is 3

Answer 1

Answer: (C)

is 2

Answer 2

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