On a Series of Goldbach and Euler
| Autor: | Lluís Bibiloni, Jaume Paradís, Pelegrí Viader |
|---|---|
| Rok vydání: | 2006 |
| Předmět: |
Perfect power
Series (mathematics) General Mathematics Mathematics::History and Overview Divergent series jel:C00 Euler's four-square identity Algebra History of mathematics infinite series nonstandard analysis symbols.namesake Simple (abstract algebra) History of mathematics Goldbach's conjecture Calculus Euler's formula symbols Applied mathematics Mathematics |
| Zdroj: | Scopus-Elsevier |
| ISSN: | 1930-0972 0002-9890 |
| DOI: | 10.1080/00029890.2006.11920299 |
| Popis: | 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 different) 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. |
| Databáze: | OpenAIRE |
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