My point is, you get the same thing with (1-x^2)^n or exp(-x*x/2) when raised to the n - the normal curve.

So to me, it seems like a product version of the central limit theorem - we keep reapplying by multiplying yet again by something roughly a normal distribution, and converges (nicely!) to the real normal distribution.

Turning the above sentence into math .. ie. stating why in the limit the repeated product converges to a scaled normal distribution… I havent done that yet - it might be obvious to others reading this blog …

Great, stimulating discussion/exposion, thanks!
[apologies for lack of latex]

Gil Kalai told me about your demystification of Stirling’s formula. Indeed, this easy result deserves to be better explained and understood, and I recall as an undergraduate I was quite frustrated at not being given a proper explanation for it. Then, about 10 years ago I discovered a new derivation of the constant sqrt(2 pi) which I believe to be slightly simpler than the one you presented, and conceptually better motivated, since it relates to the probabilistic viewpoint involving the central limit theorem. The difference with the approach you and a few of the readers mention is that I look at the cumulative probabilities of the symmetric binomial distribution instead of at just the “local” probabilities. By computing the sum of the first half of the binomial coefficients in a given row in two ways (first, using the obvious symmetry, and second, using a simple integration formula that converges to the integral of the Gaussian distribution), one gets the constant immediately.

My proof appeared in the American Math. Monthly 107 (2000), 556–557, and can be found on my web site. I wonder if you and others will think that my approach is indeed simpler than integrating powers of cosine - I find it hard to be objective about this, but of course both approaches are quite easy. By the way, I apologize for the pompous title of my paper (so pompous that I’m embarrassed to write it here) - in my defense, I was 21 when I wrote it…

In particular, on pp. 354-5 of his EXCURSIONS, Professor Young notes that summing the values of an analytic function f at the first n non-negative integers boils down to finding another function F related to it by

f(x) = F(x + 1) - F(x) = F’(x) + (1/2!)F”(x) + (1/3!)F”’(x) + … ,

since f(0) + f(1) + … + f(n-1) is then just F(n) - F(0). And now, to Euler, the striking formal resemblance between

f(x) = F’(x) + (1/2!)F”(x) + (1/3!)F”’(x) + …

and

y = x + ax^2 + bx^3 + …

naturally suggested solving for F in much the same way as one solves for x in terms of y in a power series of this sort. One simply writes

x = y + ry^2 + sy^3 + … ,

with r, s, … undetermined constants, and substitutes this form into the power series. Then one fixes r, s, … successively by requiring the coefficients of terms of like degree to match on both sides of the equation.

Of course there’s more to it than this in the end, but it’s hard to think of a more straightforward argument for putting

F’(x) = f(x) + rf’(x) + sf”(x) + … ,

and substituting back. And not only does this astonishingly simple idea actually work, it exposes the essence of the Euler-Maclaurin summation formula.

Chapter 6 (pp 109-126) is an excellent, motivated, wonderfully illustrated expository account of the path traveled largely by Gowers to Stirling’s formula. The Central Limit Theorem is discussed in the previous chapter, and so the estimate of the middle binomial coefficient 2n choose n is available. His approach to Central Limit Theorem is also excellent, starting from coin-tossing, and leading to the standard improper double integral exploited for the normal distribution. Explicit use of the big-Oh notation is avoided.

Ball also mentions that historically this approach is backwards; the Central Limit Theorem was first established using Stirling-like ideas. Also, Stirling used the Wallis product formula.

tsm [at] usna.edu

I was wondering if you’d mind if I used your blog article on Stirling’s formula for the next edition of “Eureka” (the journal of the Archimedeans undergrad. maths society here at Cambridge). It would most likely reach quite a broad audience of first years next year if you would consent. It might be worth appending David Speyer’s remarks also.

Of course if there is something else you’d prefer I’m open to suggestions.

Contact me at archim-eureka-editor@srcf.ucam.org if you are interested.