E247  De seriebus divergentibus
(On divergent series)
Summary:
One feature of this article is an evaluation of the hyergeometric series
11+26+24120+720…, by looking at the function
s(x) = x  x^{2} + 2!x^{3} –3!x^{4} + 4!x^{5} –
5!x^{6} + .... Euler attacks by interpolation methods and seems to get
the answer around 0.59. Then he integrates it and evaluates again by continued
fractions, and gets 0.5963473621372.
According to C. G. J. Jacobi, a treatise with this title was read to the Berlin Academy on October
27, 1746.
According to the records, it was presented to the St. Petersburg Academy on March 12,
1753.
Publication:

Originally published in Novi Commentarii academiae scientiarum Petropolitanae 5, 1760, pp. 205237

Opera Omnia: Series 1, Volume 14, pp. 585  617
 A handwritten French translation of this treatise can be found in the library of the observatory in Uccle, near Brussels.
Documents Available:
 Original publication: E247
 An English translation of Sections 112 of this paper can be found in Barbeau and Leah's 1976 paper (see reference below).
 Alexander Aycock has made draft translations into German and English, available online at EulerKreis Mainz.
 E247 is discussed in Ed Sandifer's How Euler Did It June 2006 column published online by the MAA.
 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 E247 include:
 Barbeau E.J., Leah, P. J., "Euler's 1760 paper on divergent series," Historia Mathematica, 3(1976), 141160
 Barbeau, E.J., "Euler subdues a very obstreporous series," The American Mathematical Monthly, 86 (1979), 356372.
 Dudley R.M., “Some inequalities for continued fractions.” Mathematics of Computation, 49 (180), pp. 585593 (Oct 1987).
 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).
 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).
 Fraser C.G., “The calculus as algebraic analysis  some observations on mathematicalanalysis in the 18thcentury.” Archive for History of Exact Sciences, 39 (4), pp. 317335 (1989).
 Golland L.A, Golland RW., “Euler troublesome series  an early example of the use of trigonometric series.” Historia Mathematica, 20 (1), pp. 5467, (Feb 1993).
 Pieper H., “On Euler contributions to the 4squares theorem.” Historia Mathematica, 20 (1), pp. 1218 (Feb 1993).
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