E61 -- De summis serierum reciprocarum ex potestatibus numerorum naturalium ortarum dissertatio altera, in qua eaedem summationes ex fonte maxime diverso derivantur
(On sums of series of reciprocals from powers of natural numbers from another discussion, in which the sums are derived principally from another source)
Euler gives an infinite product for sin(x)/x and formulas sin(x) =
(ex - e-x)/2 and cos(x) =
(ex + e-x)/2, which he had communicated
with Goldbach 9 December 1741 and 8 May 1742. He presents an infinite series for
p/sin(ps) and for
p cot(ps), and an evaluation of the sums of the reciprocals of the
odd squares as p2/8, and the alternating sum to be
According to C. G. J. Jacobi, a treatise with this title was presented to the Berlin Academy on September 6, 1742.
Originally published in Miscellanea Berolinensia 7, 1743, pp. 172-192
Opera Omnia: Series 1, Volume 14, pp. 138 - 155
- E61 can be viewed or downloaded from
Digitalisierte Akademieschriften und Schriften zur Geschichte
der Königlich Preußischen Akademie der Wissenschaften, which includes serial publications of the Prussian Academy of Science
in the 18th and 19th Centuries.
- German Translation (Alexander Aycock and Arseny Skryagin): E61
- E061 is discussed in Ed Sandifer's How Euler Did It March
2004 column published online by the MAA.
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