First slide

Introduction to limits

Question

The value of $\lim_{n \to \infty} \frac{[r] + [2 r] + . . . + [n r]}{n^{2}}$
Where $r$ is a non-zero real number and $[r]$ denotes the greatest integer less than or equal to $r$ , is equal to:

Moderate

A

$\frac{r}{2}$
B

0
C

$r$
D

$2 r$

Solution

$\begin{array}{l} Using the definition of greatest integer funciton \\ r - 1 < [r] \leq r \\ 2 r - 1 < [2 r] \leq 2 r \\ n r - 1 < [n r] \leq n r \\ \Rightarrow \frac{(r + 2 r + ...... + n r) - n}{n^{2}} < \frac{[r] + . . . + [n r]}{n^{2}} \leq \frac{(r + 2 r + . . . + n r)}{n^{2}} \\ \Rightarrow \underset{n \to \infty}{l t} \frac{r . \frac{n (n + 1)}{2} - n}{n^{2}} < \underset{n \to \infty}{l t} \frac{[r] + . . . + [n r]}{n^{2}} \leq \underset{n \to \infty}{l t} \frac{r (\frac{n (n + 1)}{2})}{n^{2}} \\ \frac{r}{2} < \underset{n \to \infty}{l t} \frac{[r] + . . . + [n r]}{n^{2}} \leq \frac{r}{2} \end{array}$
$By sandwich theorem the given limit is \frac{r}{2}$

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