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Prove the following

Question:

$\lim _{n \rightarrow \infty}\left(\frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\frac{n}{n^{2}+3^{2}}+\ldots .+\frac{1}{5 n}\right)$

  1. $\frac{\pi}{4}$

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

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

  4. $\frac{\pi}{2}$


Correct Option: , 2

Solution:

$\lim _{x \rightarrow \infty} \sum_{r=1}^{2 n} \frac{n}{n^{2}+r^{2}}$\

$\lim _{x \rightarrow \infty} \sum_{r=1}^{2 n} \frac{1}{n\left(1+\frac{r^{2}}{n^{2}}\right)}=\int_{0}^{2} \frac{d x}{1+x^{2}}=\tan ^{-1} 2$

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