E20  De summatione innumerabilium progressionum
(The summation of an innumerable progression)
Summary:
This paper talks about z(2), denoted here s_{2}=
p^{2}/6. This is about 1.644934. He says this follows from
E25 and E19,
and also refers us forward to E736. Then Euler brings in the
harmonic series: let f(x) be the xth partial sum of the harmonic series. Euler approximates this as an
integral and defines his constant g as the limit of
f(x)log(x).
According to the records, it was presented to the St. Petersburg Academy on March 5, 1731.
Publication:

Originally published in Commentarii academiae scientiarum Petropolitanae 5, 1738, pp. 91105
 Opera Omnia: Series 1, Volume 14, pp. 25  41
 Reprinted in Comment. acad. sc. Petrop. 5, ed. nova, Bononiae 1744, pp. 7588 [E20a]
Documents Available:
 Original publication: E020
(in the Commentarii)
 Ian Bruce has translated E20 into English.
 German Translation (Alexander Aycock and Arseny Skryagin): E20
 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 E20 include:
 Calinger R., "Leonhard Euler: The first St Petersburg years (17271741)." Historia Mathematica, 23 (2), pp. 121166 (May 1996).
 Ferraro G, Panza, M., "Developing into series and returning from series: A note on the foundations of eighteenthcentury analysis." Historia Mathematica, 30 (1), pp. 1746 (Feb 2003).
 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).
 McKinzie M, Tuckey C., "Hidden lemmas in Euler's summation of the reciprocals of the squares." Archive for History of Exact Sciences, 51 (1), pp. 2957 (1997).
 Sandifer E., "Estimating the Basel Problem." How Euler Did It. (Published online by the MAA.)
Return to the Euler Archive