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 p2/8√(2).

According to C. G. J. Jacobi, a treatise with this title was presented to the Berlin Academy on September 6, 1742.

Publication: Documents Available:

Return to the Euler Archive