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In Upper Limit on the Central Binomial Coefficient, Noam Elkies and David Speyer have given a nice proof that the central binomial coefficient $\binom{2n}{n} \sim \frac{4^n}{\sqrt{\pi n}}$. This can be used to derive Stirling's formula $$ a_n = \frac{n!e^n}{n^n \sqrt{n}} \sim \sqrt{2\pi} $$ by showing that $a_n$ is decreasing, hence convergent to some positive real number $c$, and computing $c$ via $$ c = \lim_{n \to \infty} \frac{a_n^2}{a_{2n}} = \sqrt{2\pi}. $$ My question is whether we can do without this roundabout tour and prove Stirling's formula directly along the lines of the proofs quoted above.

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    $\begingroup$ Why is that more slick than the Laplace/stationary phase method for the Gamma function integral which also easily gives the complete asymptotic expansion, e.g., connecting to semiclassical expansion in terms of Feynman diagrams graded by number of loops? But +1 anyway $\endgroup$ Dec 31, 2020 at 19:58
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    $\begingroup$ Terence Tao explains the Laplace's method argument here: terrytao.wordpress.com/2010/01/02/… $\endgroup$ Dec 31, 2020 at 20:54
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    $\begingroup$ For what it is worth, that asymptotic estimate in the form $\binom{2n}{n} \sim c4^n/\sqrt{2n}$ for an unknown constant $c \approx .797$ was essentially the context in which Stirling’s formula was discovered; Stirling determined that $c = \sqrt{2/\pi}$. $\endgroup$
    – KConrad
    Jan 2, 2021 at 9:09
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    $\begingroup$ I think, this two-steps argument which you refer to is natural. The limit of $a_{2n}/a_n^2$ (given by central binomials asymptotics) does not change when we multiply the sequence $a_n$ by any exponential function $A^n$. That's why it is strictly weaker than the asymptotics of $a_n$. So, it is natural that two "independent" constants $\pi$ and $e$ in Stirling formula must appear "independently". $\endgroup$ Jan 2, 2021 at 10:03
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    $\begingroup$ @JoséHdz.Stgo. see the third paragraph of Section 5 of kconrad.math.uconn.edu/blurbs/analysis/stirling.pdf. $\endgroup$
    – KConrad
    Jan 20, 2021 at 7:45

3 Answers 3

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The proof in the OP based on the sequence $a_n$ is proof number 1 in Steve Dunbar's Dozen Proofs of Stirling’s Formula (page 8, worked out here). Is there an alternative proof based on a sequence $b_n$ that does not require knowledge of Stirling's formula to construct the sequence?

Proof number 2 in that list gives such an alternative, based on $b_n=\sum_{k=1}^n\log k=\log n!$ Approximation of the sum by the integral gives $$b_n\approx \int_0^n\log x\,dx+\text{error}=n\log n -n+\text{error}.$$ The calculation of the error term, to obtain the asymptotic $\log\sqrt{2\pi n}$, can be done using the Euler-MacLaurin formula.$^\ast$ If we prefer not to use Euler-MacLaurin, proofs number 3 and 4 start from the same sequence and then approximate the sum by an integral using the trapezoidal rule or a Taylor series. These calculations are worked out in Dunbar's notes.

$^\ast$ Dunbar credits this proof to John Todd, Introduction to the Constructive Theory of Functions (1963).

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    $\begingroup$ The links in this post no longer work, but at leat the first one and the second one can be found in the Wayback Machine. $\endgroup$ Nov 26, 2023 at 8:28
  • $\begingroup$ thank you, @MartinSleziak -- I replaced the two broken links with the archived ones; too bad the last broken link has not been archived. $\endgroup$ Nov 26, 2023 at 12:17
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I've played around with this a bit. I have a slick lower bound, but not a slick upper bound.

We start with the $\Gamma$-integral: $$n! = \int_{x=0}^\infty x^n e^{-x} dx = \int_{y=-n}^\infty (n+y)^n e^{-n-y} dy$$ so $$\frac{n! e^n}{n^n} = \int_{y=-n}^\infty (1+y/n)^n e^{-y} dy. \tag{$\clubsuit$}$$

Lower bound

From $(\clubsuit)$, we have $$\frac{n! e^n}{n^n} \geq \int_{y=-n}^n (1+y/n)^n e^{-y} dy. \tag{$\diamondsuit$}$$ From the arithmetic-geometric mean inequality, $$(1-y/n)^n e^{-y} + (1+y/n)^n e^{-y} \geq 2 (1-y^2/n^2)^{n/2}$$ so, combining the values of the integrand in $(\diamondsuit)$ at $y$ and $-y$ $$\frac{n! e^n}{n^n} \geq \int_{y=-n}^n (1-y^2/n^2)^{n/2} \, dy.$$ Putting $y = n \sin \theta$, we get $$\frac{n! e^n}{n^n} \geq \int_{\theta = -\pi/2}^{\pi/2} (\cos^n \theta) (n \cos \theta) \, d \theta= n \int_{\theta = -\pi/2}^{\pi/2} \cos^{n+1} \theta \, d \theta . $$ Now, using the same trick of putting $u = \tan \theta$ as in my other answer, we get $$\int_{\theta = -\pi/2}^{\pi/2} \cos^{n+1} \theta \, d \theta = \int_{u=\infty}^\infty \frac{du}{(1+u^2)^{n/2+3/2}} \geq \int_{u=\infty}^\infty e^{-u^2 (n/2+3/2)} \, du = \frac {\sqrt{\pi}}{\sqrt{n/2+3/2}}.$$

Putting it together, $$\frac{n! e^n}{n^n} \geq \frac{\sqrt{\pi} n}{\sqrt{n/2+3/2}}$$ or $$n! \geq \frac{n^n}{e^n} \sqrt{2\pi n} \frac 1 {\sqrt{1+3/n}}.$$

Failure to get an upper bound Unfortunately, there seems to be no way to get a similar concrete upper bound for the $\Gamma$ integral. We can split $(\clubsuit)$ into $$\frac{n! e^n}{n^n} = \int_{z=0}^{n} (1-z/n)^n e^z \, dz + \int_{y=0}^\infty (1+y/n)^n e^{-y} \, dy. \qquad \tag{$\heartsuit$} $$ The first integrand is tractable: We have $$\log (1-z/n) \leq -\frac{z}{n}-\frac{z^2}{2n^2} \qquad \text{for $z>0$}$$ (since the other terms of the Taylor series are all negative) so $$(1-z/n)^n e^z \leq \exp(-z-z^2/(2n)) e^z = \exp(-z^2/(2n))$$ and $$\int_{z=0}^n (1-z/n)^n e^z \, dz \leq \int_{z=0}^n e^{-z^2/(2n)} \, dz \leq \int_{z=0}^\infty e^{-z^2/(2n)} \, dz = \frac{\sqrt{2 \pi n}} 2.$$ So the remaining task is to find a way to upper bound the second integral in $(\heartsuit)$ by $\tfrac{\sqrt{2 \pi n}} 2 (1+o(1))$. There are lots of ways to prove that this is the right asymptotic growth, but I haven't found any that yields a slick concrete inequality.


I skimmed through the many proofs of Stirling's formula that the Monthly has published in the last 70 years. When one starts from this approach, the usual route is to break the integral $\int_{y=0}^\infty (1+y/n)^n e^{-y} \, dy$ into two pieces, roughly at $y=cn$.

For $y<n$, one can use the bound $$\log (1+y/n) \leq \frac{y}{n} - \frac{y^2}{2n^2} + \frac{y^3}{3 n^3}$$ coming from the alternating series test; one then has to do something messy with the cube in the exponent.

For $y>cn$, there are a variety of approaches to show that $\int_{y=cn}^\infty (1+y/n)^n e^{-y} \, dy \to 0$ as $n \to \infty$. The one that I find simplest is to find a $b<1$ such that $1+u \leq \exp(bu)$ for $u>c$, and then we have $\int_{y=cn}^\infty (1+y/n)^n e^{-y} \, dy \leq \int_{y=cn}^\infty e^{-(1-b)y} \, dy$.

I haven't found anyone who just makes a slick change of variables and deals with the integral in one swoop.

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  • $\begingroup$ I also played with the series formula $\frac{n! e^n}{n^n} = \sum_{j=1}^{n} (1-1/n)(1-2/n) \cdots (1-(j-1)/n) + 1 +\sum_{k=1}^{\infty} \tfrac{1}{(1+1/n)(1+2/n) \cdots (1+k/n)}$, gotten by rearranging $e^n = \sum_{i=0}^{\infty} \tfrac{n^i}{i!}$: The variables relate by $i=n-j$ and $i=n+k$. It's the same situation -- you get a good lower bound from AM-GM, and a good upper bound on the first sum, but I see no good way to upper bound the second sum. $\endgroup$ Nov 8, 2023 at 17:17
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    $\begingroup$ Upper bound for the second integral: in AMM problem 11353 (solution in AMM 117 (January 2010)) it is shown that $$\frac{2}{3}<\int_{0}^\infty(1+\frac{x}{n})^n e^{-x}\,dx -\sqrt{\frac{\pi}{2}n}<1$$ $\endgroup$
    – esg
    Nov 27, 2023 at 17:23
  • $\begingroup$ @esg Very cool! I'm not sure what the ethics is of simply retyping the solution in order to evade copyright. If I can find any significant simplifications, I'll write up the AMM solution. $\endgroup$ Nov 28, 2023 at 1:58
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    $\begingroup$ Luckily, the upper bound has a rapid proof. $$I(n):=\int_0^\infty (1+\frac{x}{n})^n\,e^{-x}\,dx=n\,\int_{0}^{\infty}(1+y)^n e^{-ny}\,dy=n\int_0^\infty e^{-n\,\frac{z^2}{2}}\frac{z}{y(z)}(1+y(z))\,dz$$ where in the last step the substitution $y=y(z)$ was used, with $z(y):=\sqrt{2\,\left(y-\log(1+y)\right)}$ (and y(z) it's inverse). Now $\frac{z(y)}{y}\leq 1$, hence $I(n)\leq n\int_0^\infty e^{-n\,\frac{z^2}{2}}(1+z)\,dz=\sqrt{\frac{\pi}{2}\,n}+1\;.$ $\endgroup$
    – esg
    Dec 1, 2023 at 17:26
  • $\begingroup$ Thanks! That IS pretty rapid. $\endgroup$ Dec 1, 2023 at 20:11
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A very elementary proof of Stirling's formula is found here or the article by M. R. Murty and K. Sampath, "A very simple proof of Stirling's formula", The Mathematics Student, Vol. 84, Nos. 1-2, January-June (2015), 129–133.

This link (Wayback Machine) outlines how this proof can be done essentially in three elementary steps, with the additional assumption that we know Wallis product formula for $\pi$ (that can be proved easily by integrating $\sin ^n (x)$, see for example the Wikipedia entry). I believe this to be the most elementary proof found anywhere. It requires nothing beyond elementary Calculus.

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