Question

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

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

Answer 1

Answer: (B)

${\lim }_{n\to \infty }{\sum }_{r=1}^{2n}\frac{n}{n^2+r^2}$
${\lim }_{n\to \infty }{\sum }_{r=1}^{2n}\frac{1}{n(1+\frac{r^2}{n^2})}={\int }_0^2\frac{dx}{1+x^2}=ta{n}^{-1}2$