E43  De progressionibus harmonicis observationes
(On harmonic progressions)
Summary:
Euler gives g to six decimal places (only 5 are correct, I think he says) and gives a formal
summation that g equals ½ s_{2}  1/3 s_{3} + ¼ s_{4}
 1/5 s_{5} + .... He evaluates some other series related to log(n).
According
to the records, it was presented to the St. Petersburg Academy on March 11, 1734.
Publication:

Originally published in Commentarii academiae scientiarum Petropolitanae 7, 1740, pp. 150161

Opera Omnia: Series 1, Volume 14, pp. 87  100
 Reprinted in Comment. acad. sc. Petrop. 7, ed. nova, Bononiae 1748, pp. 138150 [E43a]
Documents Available:
 Original publication: E43
(in the Commentarii)
 Alexander Aycock has made draft translations into German and English, available online at EulerKreis 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 EulerMaclaurin summation formula.” Historia Mathematica, 25 (3), pp. 290317 (Aug 1998).
 Kline M., “Euler and infinite series.” Mathematics Magazine, 56 (5), pp. 307314 (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. 195245 (1989).
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