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  4. lim limitsn → ∞(3/n) 1+√(n/n+3)+√(n/n+6)+√(n/n+9)+....+√(n/n+3(n-1))

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Q. $ \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\}$

WBJEEWBJEE 2019

Solution:

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