lim-x-0-1-x-2-x-n-x-n-1-x-

Question Number 164702 by mathls last updated on 20/Jan/22

l i \underset{x \to 0}{m} {(\frac{1^{x} + 2^{x} + \cdot \cdot \cdot + n^{x}}{n})}^{\frac{1}{x}} = ?

Answered by mahdipoor last updated on 21/Jan/22

\lim_{x \to 0} A = \lim_{x \to 0} \frac{1}{x} l n (\frac{1^{x} + \dots + n^{x}}{n}) \Rightarrow H o p \Rightarrow

= \lim_{x \to 0} \frac{\frac{(1^{x} . l n 1 + \dots n^{x} . l n n) / n}{(1^{x} + \dots + n^{x}) / n}}{1} = \frac{(l n 1 + \dots + l n n) / n}{(1 + \dots + 1) / n}

= l n (n!) / n

\lim_{x \to 0} {(\frac{1^{x} + \dots + n^{x}}{n})}^{\frac{1}{x}} = e^{\lim_{x \to 0} A} = e^{l n (n!) / n} = {(n!)}^{1 / n}