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