26 October 2015

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