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 as well as the formulas sin(x) = (eix – e-ix)/(2i) and cos(x) = (eix + e-ix)/2, which he had communicated with Goldbach on 9 December 1741 and 8 May 1742. He presents an infinite series for π/sin(πs) and for π∙cot(πs), an evaluation of the sums of the reciprocals of odd squares as π2/8, and an evaluation of the corresponding alternating sum as π2/(8√2).
Original Source Citation
Miscellanea Berolinensia, Volume 7, pp. 172-192.
Opera Omnia Citation
Series 1, Volume 14, pp. 138-155.