Paper #776

Title:

On a series of Goldbach and Euler

Authors:

Lluís Bibiloni, Pelegrí Viader and Jaume Paradís

Date:

September 2004

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 dierent) 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

JEL codes:

C00

Area of Research:

Statistics, Econometrics and Quantitative Methods

Published in:

American Mathematical Monthly 113 (3) pp 206-220
With the title:
On a Series of Goldbach and Euler

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