Question

$\underset{n\to \infty }{\lim }\frac{1+32+243+\dots +n^5}{n^6}=$

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

Answer 1

Answer: (C)

Given,
$\underset{n\to \infty }{\lim }\frac{1+32+243+\dots +n^5}{n^6}$
$=\underset{n\to \infty }{\lim }\frac{1^5+2^5+3^5+\dots +n^5}{n^5}\cdot \frac{1}{n}$
$S=\underset{n\to \infty }{\lim }\sum _{r=1}^n{\left(\frac{r}{n}\right)}^5\cdot \frac{1}{n}$
$\Rightarrow S=\underset{0}{\overset{1}{\int }}{x}^{5}dx$
$S={\left(\frac{x^6}{6}\right)}_0^1$
$\Rightarrow S=\frac{1}{6}$