lim _{n rightarrow infty}left[frac{n+3}{n^2+1 | vedclass

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Question

(A) The given expression is $S = \lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{n+3r}{n^2+r^2}$.Dividing numerator and denominator by $n^2$, we get $

Vedclass · Accepted Answer

$\frac{\pi}{4}+\frac{3}{2} \ln 2$

Vedclass · Answer

(A) The given expression is $S = \lim _{n ightarrow \infty} \sum_{r=1}^n \frac{n+3r}{n^2+r^2}$.Dividing numerator and denominator by $n^2$, we get $S = \lim _{n ightarrow \infty} \sum_{r=1}^n \frac{\frac{1}{n} + 3\frac{r}{n^2}}{1 + (\frac{r}{n})^2} = \lim _{n ightarrow \infty} \frac{1}{n} \sum_{r=1}^n \frac{1 + 3(\frac{r}{n})}{1 + (\frac{r}{n})^2}$.Using the definition of definite integral as the limit of a sum, we have $\int_0^1 \frac{1+3x}{1+x^2} dx$.This can be split into $\int_0^1 \frac{1}{1+x^2} dx + 3 \int_0^1 \frac{x}{1+x^2} dx$.Evaluating the integrals, we get $[	an^{-1} x]_0^1 + \frac{3}{2} [\ln(1+x^2)]_0^1$.$= (\frac{\pi}{4} - 0) + \frac{3}{2} (\ln 2 - \ln 1) = \frac{\pi}{4} + \frac{3}{2} \ln 2$.

$\lim _{n \rightarrow \infty}\left[\frac{n+3}{n^2+1^2}+\frac{n+6}{n^2+2^2}+\frac{n+9}{n^2+3^2}+\ldots+\frac{2}{n}\right]=$

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