The value of

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:

  1. $\frac{1}{2} \tan ^{-1}(2)$

  2. $\frac{1}{2} \tan ^{-1}(4)$

  3. $\tan ^{-1}(4)$

  4. $\frac{1}{4} \tan ^{-1}(4)$


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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