A Tale of Two Series - William Dunham

Stony Brook Mathematics Colloquium
William Dunham, Bryn Mawr College
A Tale of Two Series
October 24, 2019

Slides: http://www.math.stonybrook.edu/Videos...

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 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 result addressed 1 + 1/4 + 1/9 + 1/16 + ... - i.e., the sum of reciprocals of the squares. Euler first evaluated this in 1734, and revisited it in 1741, but here we examine his 1755 argument 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.