27 October 2015

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.