Question

$\displaystyle\lim_{n \to \infty} \bigg(\frac{n}{n^2 + 1^2} + \frac{n}{n^2+2^2} + \frac{n}{n^2+3^2}+ ... + \frac{1}{5n}\bigg)$ is equal to :

• A. $\frac{\pi}{4}$
• B. $tan^{-1}(2)$
• C. $tan^{-1}(3)$
• D. $\frac{\pi}{2}$

Answer 1

Answer: (B)

$\lim_{n \to \infty} \sum_{r=1}^{2n} \frac{n}{n^2 + r^2}$
$\lim_{n \to \infty} \sum_{r=1}^{2n} \frac{1}{n\bigg(1 + \frac{r^2}{n^2}\bigg)} \, \, = \int_0^2 \frac{dx}{1+x^2} \, = tan^{-1}2$