Question

The value of
$ \displaystyle\lim_{n\to\infty} \left[\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots+\frac{1}{n^{2}+2n}\right] $ is

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

Answer 1

Answer: (A)

$ \displaystyle\lim_{n\to\infty} \left[\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\ldots+\frac{n}{n^{2}+n^{2}}\right]$

$=\displaystyle\lim_{n\to\infty} \sum_{r=1}^{n} \frac{n}{n^{2}+r^{2}}$

$=\displaystyle\lim_{n\to\infty} \frac{1}{n} \sum_{r=1}^{n} \frac{1}{1+\left(\frac{r}{n}\right)^{2}}$

$=\int\limits_{0}^{1} \frac{dx}{1+x^{2}}=\left[tan^{-1}\,x\right]_{0}^{1}$

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