E46 -- Methodus universalis serierum convergentium summas quam proxime inveniendi

(Universal methods of series)

Summary:

This paper begins with an expression of the trapezoid rule for "formal mechanical quadrature." Euler evaluates the first ten terms of z(2) to be 1.549768 (probably in his head) and gives an expression for the error term. Then he finds the sum of the first million terms of the harmonic series to be 14.392669.

According to the records, it was presented to the St. Petersburg Academy on June 9, 1735.

Publication:

Originally published in Commentarii academiae scientiarum Petropolitanae 8, 1741, pp. 3-9
Opera Omnia: Series 1, Volume 14, pp. 101 - 107
Reprinted in Comment. acad. sc. Petrop. 8, ed. nova, Bononiae 1752, pp. 1-6 +1 diagram [46a]

Documents Available:

Original publication: E046
A translation of E46 has been prepared by Thomas Osler and Walter Jacob, IV.
Osler and Jacob have also prepared a section-by-section Synopsis of E46. This is a good document to read for those looking for just the main ideas this work.
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 E46 include:
- Apostol TM., “An elementary view of Euler's summation formula.” American Mathematical Monthly, 106 (5), pp. 409-418 (May 1999).
- 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).
- Grabiner JV., “Was Newton's calculus a dead end? The continental influence of Maclaurin's treatise of fluxions.” American Mathematical Monthly, 104 (5), pp. 393-410 (May 1997).
- 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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