Question

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

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

Answer 1

Answer: (A)

We have $\displaystyle \lim_{n \to\infty} \frac{1^{p} + 2^{p} +....+n^{p}}{n^{p+1}}; $
$\displaystyle \lim_{n \to\infty} \sum^{n}_{r=1} \frac{r^{p}}{n^{p}.n} = \int \limits^{1}_{0} x^{p}dx = \left[\frac{x^{p+1}}{p+1}\right]_{0}^{1} = \frac{1}{p+1} $