The correct option is C 2/3 lt_n→ #10;∞∑_r=1^n1n√(rn) #10; =∫_0^1√(x) dx=23x^^3/_2|_0^1 #10; =2/3 ...

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Definite Integral as Limit of Sum

nLt→∞ √1+√2...

Question

$n L t \to \infty$ ${\frac{\sqrt{1} + \sqrt{2} + \sqrt{3} + \dots . + \sqrt{n}}{n \sqrt{n}}} =$

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2/3

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3/2

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Solution

The correct option is C 2/3
$l t_{n \to \infty} \sum_{r = 1}^{n} \frac{1}{n} \sqrt{\frac{r}{n}}$
$= \int_{0}^{1} \sqrt{x} d x = \frac{2}{3} x^{^{3} /_{2}} |_{0}^{1}$
$= 2 / 3$

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