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.
Originally published in Commentarii academiae scientiarum Petropolitanae 7, 1740, pp. 123-134
Opera Omnia: Series 1, Volume 14, pp. 73 - 86
- Reprinted in Comment. acad. sc. Petrop. 7, ed. nova, Bononiae 1748, pp. 112-123 + 1 diagram
- Original publication: E041
(in the Commentarii)
- English translation (Jordan Bell): E41
- Ian Bruce has made both a
translation and transcription of E41 available at his page Mathematical Works of the 17th Century.
- The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E41 include:
- Apostol TM., “Another elementary proof of Euler's formula for zeta(2N).” American Mathematical Society, 80 (4), pp. 425-431 (1973).
- Ferraro G., “Some aspects of Euler's theory of series: Inexplicable functions and the Euler-Maclaurin summation formula.” Historia Mathematica, 25 (3), pp. 290-317 (Aug 1998).
- Hofmann JE., “Bernoulli, Johann Cycle rectification caused by involutive formation.” Centaurus, 29 (2), pp. 89-99 (1986).
- Katz VJ., “The calculus of the trigonometric functions.” Historia Mathematica, 14 (4), pp. 311-324 (Nov 1987).
- Kline M., “Euler and infinite series.” Mathematics Magazine, 56 (5), pp. 307-314 (1983).
- Sandifer E., "The Basel Problem with integrals." How Euler Did It. (Published online
by the MAA.) Sandifer gives a thorough treatment of Euler's solution to the Basel
Problem from both E63 and E41.
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