On a series of Goldbach and Euler

Lluís Bibiloni, Pelegrí Viader (pelegri.viader@upf.edu) and Jaume Paradís (jaume.paradis@upf.edu)
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Pelegrí Viader: https://www.upf.edu/web/econ/faculty/-/asset_publisher/6aWmmXf28uXT/persona/id/3418630

Economics Working Papers from Department of Economics and Business, Universitat Pompeu Fabra

Abstract: Theorem 1 of Euler’s paper of 1737 'Variae Observationes Circa Series Infinitas', states the astonishing result that the series of all unit fractions whose denominators are perfect powers of integers minus unity has sum one. Euler attributes the Theorem to Goldbach. The proof is one of those examples of misuse of divergent series to obtain correct results so frequent during the seventeenth and eighteenth centuries. We examine this proof closely and, with the help of some insight provided by a modern (and completely di erent) proof of the Goldbach-Euler Theorem, we present a rational reconstruction in terms which could be considered rigorous by modern Weierstrassian standards. At the same time, with a few ideas borrowed from nonstandard analysis we see how the same reconstruction can be also be considered rigorous by modern Robinsonian standards. This last approach, though, is completely in tune with Goldbach and Euler’s proof. We hope to convince the reader then how, a few simple ideas from nonstandard analysis, vindicate Euler's work.

Keywords: History of mathematics; infinite series; nonstandard analysis (search for similar items in EconPapers)
JEL-codes: C00 (search for similar items in EconPapers)
Date: 2004-09
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