Question

$\displaystyle\lim _{n \rightarrow \infty} \frac{1+32+243+\ldots+n^{5}}{n^{6}}=$

• A. $\frac{1}{5}$
• B. $\frac{1}{11}$
• C. $\frac{1}{6}$
• D. $\frac{1}{\text { ? }}$

Answer 1

Answer: (C)

Given,
$\displaystyle\lim _{n \rightarrow \infty} \frac{1+32+243+\ldots+n^{5}}{n^{6}}$
$=\displaystyle\lim _{n \rightarrow \infty} \frac{1^{5}+2^{5}+3^{5}+\ldots+n^{5}}{n^{5}} \cdot \frac{1}{n}$
$S=\displaystyle\lim _{n \rightarrow \infty} \displaystyle\sum_{r=1}^{n}\left(\frac{r}{n}\right)^{5} \cdot \frac{1}{n} $
$\Rightarrow S=\int\limits_{0}^{1} x^{5} d x$
$S=\left(\frac{x^{6}}{6}\right)_{0}^{1}$
$ \Rightarrow S=\frac{1}{6}$