Question

${lim}_{n\to \infty }\frac{1^p+2^p+3^p+...+n^p}{n^{p+1}}$ is

• A. $\frac{1}{p+1}$
• B. $\frac{!}{1-p}$
• C. $\frac{1}{p}-\frac{1}{p-1}$
• D. $\frac{1}{p+2}$

Answer 1

The correct option is A $\frac{1}{p+1}$

Given,

${lim}_{n\to \infty }\frac{1^p+2^p+3^p+..........+n^p}{n^{p+1}}$

${lim}_{n\to \infty }\frac{{\sum }_{r=1}^n{r}^p}{n^p\cdot n}$

${lim}_{n\to \infty }{\sum }_{r=1}^n\left(\frac{r^p}{n^p}\right)\cdot \frac{1}{n}$

as $\frac{r}{n}=x\to \frac{1}{n}=dx$

$={\int }_0^1{n}^{p}dx$

$={\left[\frac{n^{p+1}}{p+1}\right]}_0^1$

$=\frac{1}{p+1}$