On a series of Goldbach and Euler
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- dc.contributor.author Bibiloni, Lluísca
- dc.contributor.author Viader Canals, Pelegríca
- dc.contributor.author Paradís, Jaumeca
- dc.contributor.other Universitat Pompeu Fabra. Departament d'Economia i Empresa
- dc.date.accessioned 2017-07-26T10:50:41Z
- dc.date.available 2017-07-26T10:50:41Z
- dc.date.issued 2004-09-01
- dc.date.modified 2017-07-23T02:08:44Z
- dc.description.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.
- dc.format.mimetype application/pdfca
- dc.identifier https://econ-papers.upf.edu/ca/paper.php?id=776
- dc.identifier.citation American Mathematical Monthly 113 (3) pp 206-220
- dc.identifier.uri http://hdl.handle.net/10230/382
- dc.language.iso eng
- dc.relation.ispartofseries Economics and Business Working Papers Series; 776
- dc.rights L'accés als continguts d'aquest document queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons
- dc.rights.accessRights info:eu-repo/semantics/openAccess
- dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/3.0/es/
- dc.subject.keyword history of mathematics
- dc.subject.keyword infinite series
- dc.subject.keyword nonstandard analysis
- dc.subject.keyword Statistics, Econometrics and Quantitative Methods
- dc.title On a series of Goldbach and Eulerca
- dc.title.alternative On a Series of Goldbach and Euler
- dc.type info:eu-repo/semantics/workingPaper