JEE PYQ: Limits Question 3
Question 3 - 2021 (17 Mar Shift 2)
The value of $\lim_{n \to \infty} \frac{[r] + [2r] + \cdots + [nr]}{n^2}$, where $r$ is non-zero real number and $[r]$ denotes the greatest integer less than or equal to $r$, is equal to:
(1) $\frac{r}{2}$
(2) $r$
(3) $2r$
(4) $0$
Show Answer
Answer: (1)
Solution
We know $kr - 1 < [kr] \leq kr$. Summing: $\frac{n(n+1) \cdot r}{2 \cdot n^2} \to \frac{r}{2}$ and $\frac{n(n+1)r/(2) + n}{n^2} \to \frac{r}{2}$. By Sandwich Theorem, $\lim_{n\to\infty} \frac{[r]+[2r]+\cdots+[nr]}{n^2} = \frac{r}{2}$.
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