Seminars & Colloquia Calendar
A Tale of Two Series
William Dunham, Bryn Mawr College
Location: HILL 705
Date & time: Wednesday, 11 October 2017 at 3:30PM - 4:30PM
Abstract: 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 a 1737 paper, Euler considered 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + … – i.e., the sum of reciprocals of the primes – and established that this 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 described this argument 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 derived the sum by using l’Hospital’s rule not once, not twice, but thrice! Euler has been described as “analysis incarnate.” These two proofs should leave no doubt that such a characterization is apt.
NOTE: This talk is accessible to anyone who has completed the calculus sequence.