Find the limit $\lim_{n\to \infty}\{{n!}^{1/n}\}/n$

Question

Find the limit $$\lim_{n\to \infty}\{{(n!)}^{1/n}\}/n$$

I took exp log but getting answer as 1 but it should be 1/e. Required a nice approach.

Stirling's approximation:$$n!\sim\left(\frac ne\right)^n$$This version should suffice :-) Or to be more accurate,$$n!\sim\sqrt{2\pi n}\left(\frac ne\right)^n$$ — Simply Beautiful Art, Apr 16 '17 at 15:16
https://math.stackexchange.com/questions/1598508/i-need-help-to-advance-in-the-resolution-of-that-limit-lim-n-to-infty-s?noredirect=1&lq=1 — lab bhattacharjee, Apr 16 '17 at 15:17
See also Finding the limit of $\frac {n}{\sqrt[n]{n!}}$ and other posts linked there. — Martin Sleziak, Apr 17 '17 at 06:53

score 1 · Accepted Answer · answered Apr 16 '17 at 15:17

1

For large $n$, we have $n! \simeq \exp (n \log n - n)$, Taking the limit then gives $\lim_{n\to \infty}\{{(n!)}^{1/n}/n\} = \frac1 e$

answered Apr 16 '17 at 15:17

Andreas

score 0 · Answer 2 · answered Apr 16 '17 at 15:17

0

Hint : Evaluate an assymptotical equivalent to $$n!^{1/n} = e^{\sum_{k=1}^n \log(k)/n}$$ by first finding one of $$\sum_{k=1}^n \log(k)\over n$$

answered Apr 16 '17 at 15:17

Astyx

2 Answers2