From Madhava–Leibniz to Lehmer’s Limit

Abstract

In a 1985 article in this Monthly, D. H. Lehmer considered a fascinating family of infinite series involving the central binomial coefficient, which exhibited limiting behavior involving the ubiquitous constant π in two different ways; yet it was not until 2012 that Dyson et al. gave a proof of the latter, naming it Lehmer’s limit in his honor. Here we give an elementary demonstration that both phenomena derive from simple transformations of the celebrated Madhava–Leibniz series for .