On the sums of series of reciprocals
According to Eneström, this paper was not handed in after its presentation at the St. Petersburg Academy.
Euler finds the values of Î¶(2n), where Î¶ is now named as the Riemann zeta function. He introduces the Euler numbers En. All the odd-indexed Euler numbers are zero, E0=1, E2=-1, E4=5, E6=-61, E8=1380, and they satisfy (E+1)n + (E-1)n = 0. (The Euler numbers are the coefficients of the series expansion of sech(x) and are related to Bernoulli numbers.) Among other things, this paper includes an infinite product formula for sin(x)/x.
Original Source Citation
Commentarii academiae scientiarum Petropolitanae, Volume 7, pp. 123-134.
Opera Omnia Citation
Series 1, Volume 14, pp.73-86.