Question

$\underset{n\to \infty }{lim}[\frac{1}{1+n^2}+\frac{2}{2^2+n^2}+...+\frac{n}{n^2+n^2}]$ is

• A. $log2$
• B. $2log2$
• C. $\frac{\log 2}{2}$
• D. $\frac{\log 2}{3}$

Answer 1

Answer: (C)

$\frac{\log 2}{2}$

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

$={\int }_0^1\frac{x}{1+x^2}dx$
$=\frac{1}{2}{\int }_0^1\frac{d\left(x^2\right)}{1+x^2}$

$=\frac{1}{2}log(1+x^2){|}_0^1$

$=\frac{1}{2}log\left(2\right)$