Question

Evaluate the following

$\underset{n\to \infty }{lim}\frac{1^9+2^9+3^9+\dots .+n^9}{n^{10}}$

• A. $\frac{1}{2}$
• B. $\frac{1}{5}$
• C. $\frac{1}{10}$
• D. $\frac{1}{15}$

Answer 1

The correct option is C $\frac{1}{10}$

$\underset{n\to \infty }{lim}\frac{1^9+2^9+........n^9}{n^{10}}=\underset{n\to \infty }{lim}{\sum }_{r=1}^n\frac{r^9}{n^{10}}$

$=\underset{n\to \infty }{lim}{\sum }_{r=1}^n\frac{1}{n}{\left(\frac{r}{n}\right)}^9={\int }_0^1{x}^{9}dx$

$={(\frac{x^{10}}{10}+c)}_0^1$

$=\frac{1}{10}$