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E393 -- De summis serierum numeros Bernoullianos involventium

(On the sum of series involving the Bernoulli numbers)

**Summary:**

In chapter 5 of his Calculus differentialis (E212), Euler shows
for the first time how the
sequences of coefficients that arise in z(2*n*) ,
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(2*n*) . 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.

**Publication:**
- Originally published in
*Novi Commentarii academiae scientiarum Petropolitanae* 14, 1770, pp. 129-167
*Opera Omnia*: Series 1, Volume 15, pp. 91 - 130

**Documents Available:**
- Original Document: E393
- German translation (Artur Diener and Alexander Aycock): E393

- E393 is discussed in Ed Sandifer's How Euler Did It September 2005 column published online by the MAA.

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