Mathematics

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


ANSWER

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


SOLUTION
$$\displaystyle \lim_{n\rightarrow \infty }\frac{1^{n}+2^{n}+...+n^{n}}{n^{p}}\times \frac{1}{n}$$
   $$\displaystyle =\lim_{n\rightarrow \infty }\frac{1}{n}\left [ \left ( \frac{1}{n} \right )^{p}+\left ( \frac{2}{n} \right )^{p}+...+\left ( \frac{n}{n} \right )^{p} \right ]$$
   $$\displaystyle =\lim_{n\rightarrow \infty }\sum_{r=1}^{r=n}\frac{1}{n}\left ( \frac{r}{n} \right )^{p}$$
   $$\displaystyle =\int_{0}^{1}x^{n}dx=\frac{1}{p+1}$$

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Single Correct Hard Published on 17th 09, 2020
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