Computing $\lim_{n\to\infty}\frac{\log n!}{n}$

Asked Sep 08 '21 at 18:25

Active Sep 08 '21 at 18:25

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I am trying to compute the expression

$$\limsup_{n\to \infty}|n!|^\frac{1}{n}$$

My thinking is that

$$\lim_{n\to \infty}|n!^\frac{1}{n}|=\lim_{n\to \infty}n!^\frac{1}{n}=\lim_{n\to\infty}e^\frac{\log n!}{n} = e^{\lim_{n\to\infty}\frac{\log n!}{n}}$$

Then I must compute $\lim_{n\to\infty}\frac{\log n!}{n}$.

From here, I am not sure where to go. How can I compute this limit?

asked Sep 08 '21 at 18:25

random0620

$\log(n!)=\log(2)+\log(3)+\cdots+\log(n-1)+\log(n)$ – pshmath0 Sep 08 '21 at 18:31
Here is a post about $\lim_{n\to \infty}\frac{\log(n!)}{n}$ – Sep 08 '21 at 18:33
Using pxel comment: $\frac{log(n!)}{n}\gt \frac{log(n/2)}{2}\to \infty$ – herb steinberg Sep 08 '21 at 18:35

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