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Free Content When Euler Met l'Hôpital

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In the fifteenth chapter of his 1755 text on differential calculus, Leonhard Euler considered indeterminate expressions of the form 0/0 and provided a string of dazzling examples that make modern treatments of the subject look quite pedestrian. Some indeterminate forms he evaluated with basic algebra, some with trigonometric identities, and some with infinite series. And, as the title suggests, he rolled out the heavy artillery of l'Hospital's rule. In Euler's hands, this served not just to solve specially-concocted problems but to evaluate the sum of the first n whole numbers. More improbably, he applied l'Hospital's rule (three times!) to resolve the Basel problem, i.e., to sum the infinite series of the reciprocals of the squares. By examining these results, we hope to reveal a master at work.

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Document Type: Research Article

Publication date: 2009-02-01

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