Bi-Co Mathematics Colloquiuim with Dr. William Dunham

Abstract:

Leonhard Euler (1707 – 1783) was a towering figure in the history of mathematics. In this talk, we consider his treatment of l’Hospital’s rule as it appeared in his 1755 textbook Institutiones calculi differentialis. Euler began with a discussion of indeterminate forms and then introduced l’Hospital’s rule (as it was understood in the mid-18th century).

After a few neat examples, he moved to the so-called “Basel problem.” This challenge, which had been posed in 1689 by Jakob Bernoulli, sought the exact sum of 1+14⁄+19+⋯+1𝑘2+⋯ ⁄⁄. We’ll see how Euler found the answer by developing a strange formula and applying l’Hospital’s rule not once, not twice, but thrice. This spectacular argument shows why Euler has rightly been called “the high priest of sum worship.”

NOTE: The only prerequisites for the talk are differential calculus and an interest in the history of mathematics.