Question

If [.] denotes the greatest integer function, then $\underset{n\to \infty }{lim}\frac{\left[x\right]+\left[2x\right]+\left[3x\right]+....+\left[nx\right]}{n^2}$ is

• A. $0$
• B. $x$
• C. $\frac{x}{2}$
• D. $\frac{x^2}{2}$

Answer 1

Answer: (C)

$\frac{x}{2}$

$\Rightarrow li{m}_{n\to \infty }\frac{\left[x\right]+\left[2x\right]++\left[3x\right]+......+\left[nx\right]}{n^2}$

$\Rightarrow li{m}_{n\to \infty }\frac{x+2x+.......+nx+\left\{x\right\}+\left\{2x\right\}+.........\left\{nx\right\}}{n^2}$

$\Rightarrow li{m}_{n\to \infty }x\left(\frac{n(n+1)}{2n^2}\right)+\frac{\left\{x\right\}+\left\{2x\right\}+........+\left\{nx\right\}}{n^2}$

As second part lies between 0 to 1, its limit value is 0 $\left[\left(\frac{something}{\infty }\right)\right]=0$

$\Rightarrow li{m}_{n\to \infty }x(\frac{1}{2}+\frac{1}{2n})$

$\Rightarrow \frac{x}{2}$