E43 -- De progressionibus harmonicis observationes
(On harmonic progressions)
Euler gives g to six decimal places (only 5 are correct, I think he says) and gives a formal
summation that g equals ½ s2 - 1/3 s3 + ¼ s4
- 1/5 s5 + .... He evaluates some other series related to log(n).
to the records, it was presented to the St. Petersburg Academy on March 11, 1734.
Originally published in Commentarii academiae scientiarum Petropolitanae 7, 1740, pp. 150-161
Opera Omnia: Series 1, Volume 14, pp. 87 - 100
- Reprinted in Comment. acad. sc. Petrop. 7, ed. nova, Bononiae 1748, pp. 138-150 [E43a]
- Original publication: E43
(in the Commentarii)
- Alexander Aycock has made draft translations into German and English, available online at Euler-Kreis Mainz.
- 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 E43 include:
- 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).
- Kline M., “Euler and infinite series.” Mathematics Magazine, 56 (5), pp. 307-314 (1983).
- Laugwitz D., “Definite values of infinite sums - aspects of the foundations of infinitesimal analysis around 1820.” Archive for History of Exact Sciences, 39 (3), pp. 195-245 (1989).
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