lim _{n} {rightarrow infty}left(frac{n}{n^2+1 | vedclass

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Question

(C) The given expression is $\lim _{n \rightarrow \infty} \sum_{r=1}^{2 n} \frac{n}{n^2+r^2}$.Dividing the numerator and denominator by $n^2$,we get $

Vedclass · Accepted Answer

$	an ^{-1} 2$

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(C) The given expression is $\lim _{n ightarrow \infty} \sum_{r=1}^{2 n} \frac{n}{n^2+r^2}$.Dividing the numerator and denominator by $n^2$,we get $\lim _{n ightarrow \infty} \sum_{r=1}^{2 n} \frac{1}{n} \left(\frac{1}{1+(r/n)^2}ight)$.Using the definition of a definite integral as the limit of a sum,$\lim _{n ightarrow \infty} \sum_{r=1}^{kn} \frac{1}{n} f(r/n) = \int_0^k f(x) dx$.Here,$f(x) = \frac{1}{1+x^2}$ and $k = 2$.Thus,the integral is $\int_0^2 \frac{1}{1+x^2} dx$.Evaluating the integral,we get $[	an ^{-1}(x)]_0^2 = 	an ^{-1}(2) - 	an ^{-1}(0) = 	an ^{-1}(2)$.

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

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