Famous limits, 2. de Moivre - Stirling

1. de Moivre

In 1730 Abraham de Moivre published  

to which in 1733 he added a Latin supplement

(Only very few copies are extant, grab one here or here.) Five years later, he included his own English translation of this Latin paper in the 1738 edition of The Doctrine of Chances, and there we can read

Sure enough, the date of 1733-12=1721 is expressly displayed in Miscellanea, where the above-mentioned problem is treated.

What de Moivre gives amounts to:

2. Stirling joins in

 

De Moivre continues his English translation by saying that his worthy and learned friend Mr. James Stirling found that  

a result of which he admits the singular elegancy. Stirling dropped this result casually and without any proof in his 1730 Methodus Differentialis (here) in Exemplum 2 after Propositio 28:

(logarithmo circumferentiae Circuli cujus Radius est Unitas = (to) the logarithm of the circumference of the circle whose radius is unity). De Moivre established it for himself, and delivered the finished result:

As many have remarked: this formula should be called  

de Moivre-Stirling formula

and not Stirling's formula.

3. Two-sided estimates

As with Wallis's formula, we'll give a totally elementary proof of a two-sided estimate, viz.

valid for n=1,2,... For n growing to infinity, the upper estimate decreases to the lower estimate, and we obtain de Moivre-Stirling's formula as a limit.

On the blackboard below we first deduce the estimates à la de Moivre with the constant left undetermined.

And here, at last, is Stirling's constant. We use Wallis's formula in the form given on the last line of this page.

Famous limits, 1. Wallis

In this book from 1656 (consult it here) John Wallis considered, among other things, an interpolation problem which led him to consider numbers of the form

 In Proposition 191, wanting to obtain their limit, he finds what we would write as

This is Wallis's product, rightly famous. More precisely we have for any n=1,2,... the two-sided estimate

of which  Wallis's product is the limiting case. (The lower estimate increases to the upper estimate for n growing to infinity.) Our modern, very elementary, proof of these estimates relies on the integrals

They are obtained by a recursion which starts with integrating by parts. Actually, Wallis's reasoning (though lacking our modern techniques) is not unlike ours; his table reproduced above displays our very distinction in even (pares) and odd (impares). Anyhow, the two-sided estimates can be proved in no time, see blackboard below. For completeness, we even included the integrals required.

Remark. The inequalities last obtained on the blackboard can also be rearranged into

which gives

These estimates are useful in assessing the behaviour of the coefficients in

We learn from them that