E393 -- De summis serierum numeros Bernoullianos involventium

(On the sum of series involving the Bernoulli numbers)


In chapter 5 of his Calculus differentialis (E212), Euler shows for the first time how the sequences of coefficients that arise in z(2n) , the Euler-Maclaurin formula and the Taylor series expansions of certain trigonometric functions are all related to the coefficients that Jakob Bernoulli had discovered in his book on probability, Ars Conjectandi. There, Euler named these numbers the "Bernoulli numbers" and showed how all these different coefficients are related.

In E-393, Euler returns to these connections and snows how the Bernoulli numbers are related to z(2n) . He also shows how the Bernoulli numbers arise in certain integrals and uses them to give some recurrence relations on Bernoulli numbers.

According to the records, it was presented to the St. Petersburg Academy on August 18, 1768.

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