Question

$\underset{x\to \infty }{lim}\frac{1^p+2^p+3^p+\dots \dots +n^p}{n^{p+1}}$ is

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

Answer 1

The correct option is A

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

We have $\underset{x\to \infty }{lim}\frac{1^p+2^p+\dots \dots +n^p}{n^{p+1}};$

$\underset{x\to \infty }{lim}{\sum }_{r=1}^n\frac{r^p}{n^p.n}$ $={\int }_0^1{x}^{p}dx={\left[\begin{array}{c}\frac{x^{p+1}}{p+1}\end{array}\right]}_0^1=\frac{1}{p+1}$