The relationship, which I intend to develop here, looks at the sums of these

1Aug 1, 2006Translation with notes of Euler’s paperRemarques sur un beau rapport entre les series des puissances tantdirectes que reciproques Originally published in Memoires de l'academie des sciences de Berlin 17, 1768, pp. 83-106Translated by Lucas Willis and Thomas J Osler Mathematics DepartmentRowan UniversityGlassboro, NJ 08028Osler@rowan.eduIntroduction to the translation and notes:This translation is the result of a happy collaboration between student and professor. Lucas Willisis an undergraduate Mathematics Major and a French Minor. Tom Osler has been a mathematics professorfor 45 years. Together we struggled to understand this brilliant work.When translating Euler’s words, we tried to imagine how he would have written had he beenfluent in modern English and familiar with today’s mathematical jargon. Often he used very long sentences,and we frequently converted these to several shorter ones. However, in almost all cases we kept his originalnotation, even though some is very dated. We thought this added to the charm of the paper. One exceptionis Euler’s use of lx for our log x, the natural logarithm. We thought lx was too confusing.. Euler was very careful in proof reading his work, and we found few typos. When we found anerror, we called attention to it in parenthesis and italics in the body of the translation. Other errors areprobably ours. When half the translation was completed, we learned that Professor Robert Stein had made ...