lim-n-1-n-1-2-1-2-n-1-n-

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

{l i m}_{n \to \infty} \frac{1}{n} (1 + 2^{\frac{1}{2}} + \dots \dots + 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} = {(\frac{1}{\frac{1}{n}})}^{\frac{1}{n}}

1 < \lim_{n \to \infty} S_{n} < \lim_{n \to \infty} {(\frac{1}{\frac{1}{n}})}^{\frac{1}{n}} = \lim_{x \to 0} \frac{1}{x^{x}} = \frac{1}{1} = 1

\Rightarrow \lim_{n \to \infty} S_{n} = 1

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