lim _{n} {rightarrow infty}left(frac{1}{1+n^5 | vedclass

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(D) The given expression is $\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{r^4}{r^5+n^5}$.Dividing the numerator and denominator of the general term

Vedclass · Accepted Answer

$\log \sqrt[5]{2}$

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(D) The given expression is $\lim _{n \rightarrow \infty} \sum_{r=1}^n \frac{r^4}{r^5+n^5}$.Dividing the numerator and denominator of the general term by $n^5$,we get:$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{r=1}^n \frac{(\frac{r}{n})^4}{(\frac{r}{n})^5+1}$.This is a Riemann sum which can be expressed as the definite integral:$\int_0^1 \frac{x^4}{1+x^5} dx$.Let $u = 1+x^5$,then $du = 5x^4 dx$,or $x^4 dx = \frac{du}{5}$.When $x=0, u=1$ and when $x=1, u=2$.The integral becomes $\frac{1}{5} \int_1^2 \frac{1}{u} du = \frac{1}{5} [\ln |u|]_1^2 = \frac{1}{5} \ln 2$.Using the property $a \ln b = \ln b^a$,we get $\ln 2^{1/5} = \ln \sqrt[5]{2}$.

$\lim _{n}$ ${\rightarrow \infty}\left(\frac{1}{1+n^5}+\frac{2^4}{2^5+n^5}+\frac{3^4}{3^5+n^5}+\ldots+\frac{n^4}{n^5+n^5}\right)=$

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