Question

The value of
$\underset{n\to \infty }{\lim }\left[\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\dots +\frac{1}{n^2+2n}\right]$ is

• A. $\frac{\pi }{4}$
• B. $\log 2$
• C. $0$
• D. $1$

Answer 1

Answer: (A)

$\underset{n\to \infty }{\lim }\left[\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\dots +\frac{n}{n^2+n^2}\right]$

$=\underset{n\to \infty }{\lim }\sum _{r=1}^n\frac{n}{n^2+r^2}$

$=\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{r=1}^n\frac{1}{1+{\left(\frac{r}{n}\right)}^2}$

$=\underset{0}{\overset{1}{\int }}\frac{dx}{1+x^2}={\left[ta{n}^{-1}x\right]}_0^1$

$=\frac{\pi }{4}$