E41 -- De summis serierum reciprocarum

(On the sums of series of reciprocals)


Euler finds the values of z(2n). 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. They are in the coefficients of the Maclaurin series expansion of sec(x) and are related to Bernoulli numbers. A number of other series ensue, along with an infinite product for sin(x)/x.

According to the records, it was read in the St. Petersburg Academy on December 5, 1735, but not handed in.

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