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What is the value of $$\displaystyle{\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^n[\sqrt\frac{4i}{n}]}$$

Where [x] denotes the greatest integer less than or equal to x

Answer is given as $3$ but I think answer will be $3/4$ (by breaking the sum from $1$ to $n/4$ and from $n/4$ to $n$). Please clarify if it is a printing mistake or if I am missing something.

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Can you see a Riemann sum in it? –  Daniel Fischer Jun 25 at 10:08
    
This is [x]. Obviously you do not mean a Riemann sum here. –  Arghya Chakraborty Jun 25 at 10:09
    
I do mean a Riemann sum. –  Daniel Fischer Jun 25 at 10:10
    
Sorry, I don't know much about Riemann sum. Please solve it. –  Arghya Chakraborty Jun 25 at 10:11
    
If you look again at the answer to your previous question, do you see how this fits in? –  Daniel Fischer Jun 25 at 10:13

1 Answer 1

up vote 3 down vote accepted

This is nothing but the evaluation of the integral $$\int_0^1\sqrt{[4x]} dx=\int_0^{\frac{1}{4}} 0dx+\int_{\frac{1}{4}}^1 dx=\frac{3}{4}$$

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