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

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Question

(A) Let $S_n = \sum_{r=1}^n \frac{n}{n^2+r^2}$.We can rewrite the expression as $S_n = \sum_{r=1}^n \frac{n}{n^2(1 + \frac{r^2}{n^2})} = \sum_{r=1}^n

Vedclass · Accepted Answer

$\frac{\pi}{4}$

Vedclass · Answer

(A) Let $S_n = \sum_{r=1}^n \frac{n}{n^2+r^2}$.We can rewrite the expression as $S_n = \sum_{r=1}^n \frac{n}{n^2(1 + \frac{r^2}{n^2})} = \sum_{r=1}^n \frac{1}{n} \cdot \frac{1}{1 + (\frac{r}{n})^2}$.Taking the limit as $n ightarrow \infty$, we use the definition of a definite integral as the limit of a Riemann sum:$\lim _{n ightarrow \infty} \frac{1}{n} \sum_{r=1}^n f(\frac{r}{n}) = \int_0^1 f(x) dx$.Here, $f(x) = \frac{1}{1+x^2}$.Thus, $S = \int_0^1 \frac{1}{1+x^2} dx$.Evaluating the integral, we get $S = [	an^{-1} x]_0^1 = 	an^{-1}(1) - 	an^{-1}(0) = \frac{\pi}{4} - 0 = \frac{\pi}{4}$.

$\lim _{n \rightarrow \infty}\left[\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\ldots+\frac{n}{n^2+n^2}\right]=$

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