Question

$li{m}_{\pi \to \infty }\frac{1^{\rho }+2^{\rho }+3^{\rho }+\cdots +n^{\rho }}{n^{\rho +1}}=$ [AIEEE 2002]

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

Answer 1

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

$li{m}_{\pi \to \infty }\frac{1^{\rho }+2^{\rho }+3^{\rho }+\cdots +n^{\rho }}{n^{\rho +1}}=li{m}_{\pi \to \infty }{\sum }_{r=1}^n\left[\frac{r^{\rho }}{n^{\rho +1}}\right]$
$=li{m}_{\pi \to \infty }\frac{1}{n}{\sum }_{r=1}^n{\left(\frac{r}{n}\right)}^{\rho }={\int }_0^1{x}^{\rho }dx={\left[\frac{x^{\rho +1}}{p+1}\right]}_0^1=\frac{1}{p+1}$