William Dunham (Ph.D., The Ohio State University) is a historian of mathematics who has written four books on the subject: Journey Through Genius (1990), The Mathematical Universe (1994), Euler: The Master of Us All (1999), and The Calculus Gallery (2005). He co-edited, along with Don Albers and Jerry Alexanderson, an anthology titled The G. H. Hardy Reader (2015). Dunham was the MAA’s George Pólya Lecturer from 2014 to 2016 and is featured in the Teaching Company’s DVD course, “Great Thinkers, Great Theorems.”

After his retirement as Koehler Professor of Mathematics at Muhlenberg College (emeritus, 2014), Dunham has held visiting positions at Harvard, Princeton, Cornell, the University of Pennsylvania, and Bryn Mawr College. This term, he is the Paul R. and Virginia P. Halmos Endowed Visiting Professor at Santa Clara University.

Abstract: Leonhard Euler (1707 – 1783) is a towering figure from the history of mathematics. Here we look at two results that show how he acquired his lofty reputation.

In 1737, Euler considered the infinite series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ... — i.e., the sum of reciprocals of the primes — and established that the sum “is infinite.” The proof rested upon his famous product-sum formula and required a host of analytic manipulations so typical of Euler’s work. André Weil regarded this paper as marking “the birth of analytic number theory.”

The other series is 1 + 1/4 + 1/9 + 1/16 + ... — i.e., the sum of reciprocals of the squares. Euler first evaluated it in 1734 and revisited it in 1741, but we examine his argument from 1755 that summed the series by using l’Hospital’s rule not once, not twice, but thrice!

Euler has been described as “analysis incarnate.” These two series, it is hoped, will leave no doubt that such a characterization is apt.