On Wallis' formula

Abstract

In the course of mathematical progress new truths are discovered while older ones are sometimes more precisely articulated and often generalised. Because of their elegance and simplicity, however, some classical statements have been left unchanged. As an example, I have in mind the celebrated formula of John Wallis,which for more than a century has been quoted by writers of textbooks. Usually this formula is written asIn this note it is shown that ¼ < θ < ½. Unquestionably, inequalities similar to this one can be improved indefinitely but at a sacrifice of simplicity, which is why they have survived so long.