E55  Methodus universalis series summandi ulterius promota
(Universal method for summation of series, further developed)
Summary:
This, along with E617, looks at the formal series f(a)f(a+h)+f(a+2h)f(a+3h)+f(a+4h).... He apparently looks at this also in E352, as well as in E617. He looks at this through a power series, f(0) + f(1)x + f(2)x^{2} + f(3)x^{3} + ..., which is obviously what he’s looking for in the case a=0, h=1 and x=1. He does Taylor series expansions at every value, 0, 1, 2,... , sums them all, and combines terms and gets an answer that reminds Faber of E352.
According to
the records, it was presented to the St. Petersburg Academy on September 17, 1736.
Publication:

Originally published in Commentarii academiae scientiarum Petropolitanae 8, 1741, pp. 147158

Opera Omnia: Series 1, Volume 14, pp. 124  137
 Reprinted in Comment. acad. sc. Petrop. 8, ed. nova, Bononiae 1752, pp. 132144 [55a]
Documents Available:
 Original publication: E055 (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 E55 include:
 Apostol TM., “An elementary view of Euler's summation formula.” American Mathematical Monthly, 106 (5), pp. 409418 (May 1999).
 Ferraro G., “Differentials and differential coefficients in the Eulerian foundations of the calculus.” Historia Mathematica, 31 (1), pp. 3461 (Feb 2004).
 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).
 Mills S., “The independent derivations by Euler, Leonhard and Maclaurin, Colin of the EulerMaclaurin summation formula.” Archive for History of Exact Sciences, 33 (13), pp. 113 (1985).
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