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Question Number 143156 by mohammad17 last updated on 10/Jun/21

$${lim}_{n\to \mathrm{\infty }}\frac{1}{n}(1+2^{\frac{1}{2}}+......+n^{\frac{1}{n}})?$$

Answered by mr W last updated on 10/Jun/21

$$1=\frac{n\times 1}{n}<S_n<\frac{n\times n^{\frac{1}{n}}}{n}={\left(\frac{1}{\frac{1}{n}}\right)}^{\frac{1}{n}}$$$$1<\underset{n\to \mathrm{\infty }}{\lim }{S}_n<\underset{n\to \mathrm{\infty }}{\lim }{\left(\frac{1}{\frac{1}{n}}\right)}^{\frac{1}{n}}=\underset{x\to 0}{\lim }\frac{1}{x^x}=\frac{1}{1}=1$$$$\Rightarrow \underset{n\to \mathrm{\infty }}{\lim }{S}_n=1$$

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