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

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Question

(B) The given expression is $S = \lim _{n \rightarrow \infty} \sum_{r=0}^{2n} \frac{1}{n+r}$.We can rewrite this as $S = \lim _{n \rightarrow \infty}

Vedclass · Accepted Answer

$\log 3$

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(B) The given expression is $S = \lim _{n \rightarrow \infty} \sum_{r=0}^{2n} \frac{1}{n+r}$.We can rewrite this as $S = \lim _{n \rightarrow \infty} \sum_{r=0}^{2n} \frac{1}{n(1 + \frac{r}{n})}$.This is a Riemann sum of the form $\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=0}^{2n} f(\frac{r}{n})$,which equals $\int_{0}^{2} f(x) dx$.Here,$f(x) = \frac{1}{1+x}$.Therefore,$S = \int_{0}^{2} \frac{1}{1+x} dx$.Evaluating the integral,we get $S = [\ln(1+x)]_{0}^{2}$.$S = \ln(1+2) - \ln(1+0) = \ln(3) - \ln(1) = \ln(3) - 0 = \ln(3)$.

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

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