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Suppose we want to show that $$ n! \sim \sqrt{2 \pi} n^{n+(1/2)}e^{-n}$$

Instead we could show that $$\lim_{n \to \infty} \frac{n!}{n^{n+(1/2)}e^{-n}} = C$$ where $C$ is a constant. Maybe $C = \sqrt{2 \pi}$.

What is a good way of doing this? Could we use L'Hopital's Rule? Or maybe take the log of both sides (e.g., compute the limit of the log of the quantity)? So for example do the following $$\lim_{n \to \infty} \log \left[\frac{n!}{n^{n+(1/2)}e^{-n}} \right] = \log C$$

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    Keith Conrad has a good explanation of this. http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CB0QFjAA&url=http%3A%2F%2Fwww.math.uconn.edu%2F~kconrad%2Fblurbs%2Fanalysis%2Fstirling.pdf&ei=Q0n7Tuk_gaC3B7PU5M8G&usg=AFQjCNFDngMkFgLhRO9M9ujPAPRv89x96Q2011-12-28
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    Stirling's formula is a pretty hefty result, so the tools involved are going to go beyond things like routine application of L'Hopital's rule, although I am sure there is a way of doing it that involves L'Hopital's rule as a step. I've just scanned the link posted by jspecter and it looks good and reasonably elementary. Another pretty elementary treatment is Terry Tao's: http://terrytao.wordpress.com/2010/01/02/254a-notes-0a-stirlings-formula/. The two treatments I've taken the time to work through both involved doing residue calculus on the gamma function.2011-12-28
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    James: are you interested in proving that C exists or in computing its value? For the former, pretty standard procedures based on an estimation of the ratio of two consecutive terms are enough.2011-12-28
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    There's a proof along this line in [The Number $\pi$](http://www.ams.org/bookstore-getitem/item=tnp) by Eymard and Lafon. (A very nice book, btw.)2011-12-31
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    There is an exercise, or a sequence of exercises, in Spivak's *Calculus* that guides you through a proof. I don't have it on hand or know the details, but I believe it goes through the Euler–Maclaurin formula.2011-12-31
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    Short elementary proofs due to [Diaconis and Freedman](http://ocw.nctu.edu.tw/course/fourier/supplement/Stirling.pdf) and to [Patin](http://ocw.nctu.edu.tw/course/fourier/supplement/Stirling-1.pdf) were published in *The American Mathematical Monthly*.2012-01-01
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    Patin's paper is already mentioned in @Byron's answer.2012-01-02
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    Dan Romik, Stirling's Approximation for n!: The Ultimate Short Proof?, at [JSTOR](http://www.jstor.org/stable/2589351).2012-01-03
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    It is a dismaying that nobody has mentioned in this thread that Abraham de Moivre was the first to show that $$\lim_{n\to\infty} \frac{n^{n+1/2}/e^n}{n!}$$ is a positive number and to calculate the number numerically, and then James Stirling showed that it is $\sqrt{2\pi}$. ${}\qquad{}$2016-01-26

9 Answers 9