How do you compute
$$\lim_{n\to\infty} \dfrac{(n!)^{1/n}}{n}\;?$$
I know that the answer is $\dfrac{1}{e}$ by plugging it into WolframAlpha, but I have no idea how to get there.
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This has been posted several times recently. I'll look for a recent example. – Thomas Andrews Apr 15 '15 at 22:40
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This is a duplicate of this question. In a comment to that question are three other originals listed. – robjohn Apr 15 '15 at 22:46
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@ThomasAndrews: nice use of that unilateral axe! I see that your example is listed in my comment to the question I linked to. – robjohn Apr 15 '15 at 22:50
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I didn't realize I had the power to unilaterally axe. I just marked it as duplicate like I have so many times before. @robjohn – Thomas Andrews Apr 15 '15 at 22:52
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@ThomasAndrews: actually, it was the Community User that closed it. See this announcement. – robjohn Apr 15 '15 at 23:14
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From DeMoivre's formula (Stirling), we have $$n! \sim C \sqrt{n} \left(\dfrac{n}e\right)^{n}$$ We hence obtain $$\dfrac{(n!)^{1/n}}{n} \sim \dfrac1e$$
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