Question

$\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{1}{\sqrt{nr}}$ is equal to

• A. $2$
• B. $1$
• C. $0$
• D. none of these

Answer 1

Answer: (A)

$2$

$\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{1}{\sqrt{nr}}=\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{1}{n\sqrt{\frac{r}{n}}}$
$=\underset{n\to \infty }{lim}\frac{1}{n}\sum _{r=1}^n\frac{1}{\sqrt{\frac{r}{n}}}={\int }_0^1\frac{1}{\sqrt{x}}dx$
$={\left[\frac{\sqrt{x}}{\frac{1}{2}}\right]}_0^1=2$