Question:
The value of $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{2 n-1} \frac{n^{2}}{n^{2}+4 r^{2}}$ is:
Correct Option: 2,
Solution:
$L=\lim _{n \rightarrow \infty} \frac{1}{n} \cdot \sum_{r=0}^{2 n-1} \frac{1}{1+4\left(\frac{r}{n}\right)^{2}}$
$\Rightarrow \mathrm{L}=\int_{0}^{2} \frac{1}{1+4 \mathrm{x}^{2}} \mathrm{dx}$
$\Rightarrow \mathrm{L}=\left.\frac{1}{2} \tan ^{-1}(2 \mathrm{x})\right|_{0} ^{2} \Rightarrow \mathrm{L}=\frac{1}{2} \tan ^{-1} 4$
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