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Question Number 210754 by mokys last updated on 18/Aug/24

Answered by Berbere last updated on 19/Aug/24

$$1+N^r\sim N^r$$$$\underset{n\to \mathrm{\infty }}{\lim }\frac{\underset{k=1}{\sum ^N}{k}^s}{N^s}=\underset{N\to \mathrm{\infty }}{\lim }\underset{k=1}{\sum ^N}\frac{1}{N}{\left(\frac{k}{N}\right)}^s={\int }_0^1{x}^s=\frac{1}{s+1}=\frac{r-s}{r}$$$$1=r-s$$$$s+1=r$$$$\underset{N\to \mathrm{\infty }}{\lim }\frac{\underset{k=1}{\sum ^N}{k}^s}{1\underset{-}{+}{N}^r}=\underset{N\to \mathrm{\infty }}{\lim }\frac{N^r}{1\underset{-}{+}{N}^r}.\frac{1+....+N^s}{N^s}=\underset{-}{+}.\frac{r-s}{r}$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{1\underset{-}{+}{x}^r}{x^r}=\underset{-}{+}1$$

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