Question

Assertion :Statement- 1 :${lim}_{n\to \infty }\frac{1^m+2^m+3^m+....n^m}{n^{m+1}}=\frac{1}{m+1}(m\ne -1)$ Reason: Statement- 2 : The above limit equals ${\int }_0^1{x}^{m}dx$

• A. Statement -1 is True, Statement -2 is True ; Statement -2 is a correct explanation for Statement -1
• B. Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1
• C. Statement -1 is True, Statement -2 is False
• D. Statement -1 is False, Statement -2 is True

Answer 1

The correct option is A Statement -1 is True, Statement -2 is True ; Statement -2 is a correct explanation for Statement -1

$\underset{n\to \infty }{lim}\frac{1^m+2^m+3^m+....n^m}{n^{m+1}}$
$\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{r^m}{n^{m+1}}=\underset{n\to \infty }{lim}[\frac{1}{n}\sum {\left(\frac{r}{n}\right)}^m]$
${\int }_0^1{x}^{m}dx={\left[\frac{x^{m+1}}{m+1}\right]}_0^1=\frac{1}{m+1}$
Ans: A