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

In the first, Euler considers the infinite series \(1/2 + 1/3 + 1/5 + 1/7 + 1/11 + \dots\), i.e., the sum of reciprocals of the primes – and establishes that the sum "is infinite." The proof from 1737 rests upon his famous product-sum formula and requires a host of analytic manipulations so typical of Euler’s work.

The other result addresses \(1 + 1/4 + 1/9 + 1/16 + \dots\), i.e., the sum of reciprocals of the squares. Euler first evaluated this in 1734, but here we examine his 1755 argument that uses l'Hospital's rule, not once, not twice, but thrice!

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

NOTE: This talk should be accessible to any mathematics major, grad student, or faculty member.

William Dunham is a historian of mathematics who has written four books on the subject: Journey Through Genius, The Mathematical Universe, Euler: The Master of Us All, and The Calculus Gallery. He recently co-edited (with Don Albers and Jerry Alexanderson) an anthology titled The G. H. Hardy Reader. Dunham is also featured in the Teaching Company’s DVD course, "Great Thinkers, Great Theorems."

After his retirement from a 22-year career at Muhlenberg College, Dunham has held visiting positions at Harvard, Princeton, Cornell, and the University of Pennsylvania. He is presently a Research Associate in Mathematics at Bryn Mawr College.