Q. $\lim_{n \rightarrow \infty}{\frac{1}{n}\ \sum_{r = 1}^{n}\frac{r}{\sqrt{n^{2} + r^{2}}}}$ is equal to
A) $1 - \sqrt{2}$
B) $\sqrt{2} - 1$
C) $\sqrt{2}$
D) $- \sqrt{2}$
Explanation
(b)
$\lim_{n \rightarrow \infty}{\frac{1}{n}\sum_{r = 1}^{n}{\frac{\left( \frac{r}{n} \right)}{\sqrt{1 + \left( \frac{r}{n} \right)^{2}}} = \frac{1}{2}\int_{0}^{1}{\frac{2x}{\sqrt{1 + x^{2}}}dx}}}$
$= \frac{1}{2}\left\lbrack \frac{\sqrt{1 + x^{2}}}{1/2} \right\rbrack_{0}^{1} = \sqrt{2} - 1$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.