Question

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

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

Answer 1

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

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

$\underset{n\to \infty }{lim}\sum _{n^{p+1}}^n\frac{t^p}{n^{p+1}}=\underset{n\to \infty }{lim}\sum _n^{t=1}\frac{t^b}{n^p}\frac{1}{n}$

$\underset{\infty \to n}{lim}\sum _n^{t=1}{t}^p={\int }_0^1{x}^{b}dx$

$\Rightarrow {\int }_0^1\left[\frac{x^{p+1}}{p+}\right]$

$\Rightarrow \frac{x^{p+}}{p+1}{|}_0^1$

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

so option (b) is correct