Question

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

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

Answer 1

The correct option is A $\frac{\pi }{4}$

$\underset{n\to \infty }{lim}[\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\cdots +\frac{n}{n^2+n^2}]$
$=\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{n}{n^2+r^2}=\underset{n\to \infty }{lim}\frac{1}{n}\sum _{r=1}^n\frac{1}{1+{\left(\frac{r}{n}\right)}^2}$
$={\int }_0^1\frac{dx}{1+x^2}={\left[{\tan }^{-1}x\right]}_0^1$
$=\frac{\pi }{4}$