The value of $\lim_{n \rightarrow \infty}\left\lbrack \frac{n}{n^{2} + 1^{2}} + \frac{n}{n^{2} + 2^{2}} + ... + \frac{n}{n^{2} + n^{2}} \right\rbrack$

Question

Dr. Abubacker · Accepted Answer

(a)
$\lim_{n ightarrow \infty}\left\lbrack \frac{n}{n^{2} + 1^{2}} + \frac{n}{n^{2} + 2^{2}} + ... + \frac{n}{n^{2} + n^{2}} ightbrack$
$= \lim_{n ightarrow \infty}{\sum_{r = 1}^{n}\frac{n}{n^{2} + r^{2}}}$
$= \lim_{n ightarrow \infty}{\frac{1}{n}\sum_{r = 1}^{n}\frac{1}{1 + \left( \frac{r}{n} ight)^{2}}}$
$= \int_{0}^{1}{\frac{dx}{1 + x^{2}} = \lbrack	an^{- 1}{x\left. \ \  ightbrack_{0}^{1} = \frac{\pi}{4}}}$

Q. The value of $\lim_{n \rightarrow \infty}\left\lbrack \frac{n}{n^{2} + 1^{2}} + \frac{n}{n^{2} + 2^{2}} + ... + \frac{n}{n^{2} + n^{2}} \right\rbrack$

Explanation

Reviewed by Dr. Abubacker

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