Evaluate the limit: lim_{n to infty} sum_{r=1 | vedclass

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(A) The given expression is $\lim_{n 	o \infty} \sum_{r=1}^{n} \frac{n}{n^2 + r^2}$.We can rewrite this as $\lim_{n 	o \infty} \sum_{r=1}^{n} \frac{

Vedclass · Accepted Answer

$\frac{\pi}{4}$

Vedclass · Answer

(A) The given expression is $\lim_{n 	o \infty} \sum_{r=1}^{n} \frac{n}{n^2 + r^2}$.We can rewrite this as $\lim_{n 	o \infty} \sum_{r=1}^{n} \frac{n}{n^2(1 + (r/n)^2)}$.This simplifies to $\lim_{n 	o \infty} \frac{1}{n} \sum_{r=1}^{n} \frac{1}{1 + (r/n)^2}$.By the definition of a definite integral,this is equivalent to $\int_{0}^{1} \frac{1}{1 + x^2} dx$.The integral of $\frac{1}{1 + x^2}$ is $	an^{-1}(x)$.Evaluating from $0$ to $1$,we get $	an^{-1}(1) - 	an^{-1}(0) = \frac{\pi}{4} - 0 = \frac{\pi}{4}$.

Evaluate the limit: $\lim_{n \to \infty} \sum_{r=1}^{n} \frac{n}{n^2 + r^2}$

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