The value of lim _{n rightarrow infty} left[ | vedclass

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(A) The given expression is $\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{n}{n^2+r^2}$.Dividing the numerator and denominator of the term inside th

Vedclass · Accepted Answer

$\frac{\pi}{4}$

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(A) The given expression is $\lim _{n ightarrow \infty} \sum_{r=1}^n \frac{n}{n^2+r^2}$.Dividing the numerator and denominator of the term inside the summation by $n^2$, we get:$\lim _{n ightarrow \infty} \sum_{r=1}^n \frac{1/n}{1+(r/n)^2}$.This is a Riemann sum of the form $\int_0^1 f(x) \, dx$ where $f(x) = \frac{1}{1+x^2}$.Thus, the limit is $\int_0^1 \frac{1}{1+x^2} \, dx$.Evaluating the integral, we get $[	an^{-1} x]_0^1 = 	an^{-1}(1) - 	an^{-1}(0) = \frac{\pi}{4} - 0 = \frac{\pi}{4}$.

The value of $\lim _{n \rightarrow \infty} \left[ \frac{n}{n^2+1^2} + \frac{n}{n^2+2^2} + \dots + \frac{n}{n^2+n^2} \right]$ is

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