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 $\pi /4$.

MSC::

• Primary 40A30
• Secondary 11B65

Acknowledgments

The author thanks the referees for their helpful comments. All numerical computation was done using the PARI-GP calculator created by C. Batut, K. Belabas, D. Bernardi, H. Cohen, and M. Olivier.

Additional information

Notes on contributors

Paul Thomas Young

PAUL THOMAS YOUNG received his Ph.D. in mathematics from Oklahoma State University in 1988, and is a Professor of Mathematics at the College of Charleston, where he has taught for the last thirty years. The longest day of his life was Pi Day, 3/14/16, a thirty-six hour day which he began playing bass in a club in Wuhan, and ended, after crossing the International Date Line, doing mathematics at his home in Charleston.