First slide

Introduction to definite integration

Question

$\underset{n \to \infty}{L t} [\frac{\sqrt{n + 1} + \sqrt{n + 2} + .... + \sqrt{n + n}}{n \sqrt{n}}]$

Moderate

A

$\frac{2 (2 \sqrt{2} - 1)}{3}$
B

$\frac{2 \sqrt{2} - 1}{3}$
C

$\frac{2 \sqrt{2} + 1}{3}$
D

None of these

Solution

$\begin{array}{l} \lim_{n \to \infty} [\sqrt{1 + \frac{1}{n}} + \sqrt{1 + \frac{2}{n} + \dots . .} + \sqrt{1 + \frac{n}{n}}] \\ \lim_{n \to \infty} \sum_{r = 1}^{n} (\sqrt{1 + \frac{r}{n}}) = \int_{0}^{1} \sqrt{1 + x} d x \\ = {(\frac{2}{3} (1 + x)^{3 / 2})}_{0}^{1} \\ = \frac{2}{3} (2^{3 / 2} - 1) \\ \frac{= 2 (2 \sqrt{2} - 1)}{3} \end{array}$

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