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Using ideas around Riemann Integration and Improper Integrals, I am looking to find $$\large\lim_{n\to\infty}\frac{\root^n \of {n!}}{n} $$ I think it is clear that the $\frac{1}{n}$ term can represent the width of each section on a partition $P_n$, which would then imply we are seeking a function on a bounded interval. Given the numerator, I am struggling to find a way to compare this to the Riemann Sum for a function.

Any hints in the right direction would be really great! Thank you

jcneek
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    See my answer at https://math.stackexchange.com/questions/4221138/is-logn-o-lognn/4221149#4221149 – Stefan Lafon Aug 14 '21 at 13:12
  • Use Stirling's formula to determine the limit. – Hans Engler Aug 14 '21 at 13:13
  • Thank you for your comments. Unfortunately I cannot use Stirling's Formula. Using Stefan's answer I have used the continuity of log to deduce that the limit is $e^{-1}$, since the limit of the log of the expression is -1, comparing it to the integral of log from 0 to 1 – jcneek Aug 14 '21 at 13:29
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    Also check out: https://www.youtube.com/watch?v=89d5f8WUf1Y – talbi Aug 14 '21 at 15:00

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