E247 -- De seriebus divergentibus
(On divergent series)
One feature of this article is an evaluation of the hyergeometric series
1-1+2-6+24-120+720-…, by looking at the function
s(x) = x - x2 + 2!x3 –3!x4 + 4!x5 –
5!x6 + .... 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
According to the records, it was presented to the St. Petersburg Academy on March 12,
Originally published in Novi Commentarii academiae scientiarum Petropolitanae 5, 1760, pp. 205-237
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.
- Original publication: E247
- A Translation of sections 1-12 of this paper can be found in Barbeau and Leah's 1976 paper (see
- 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), 141-160
- Barbeau, E.J., "Euler subdues a very obstreporous series," The American Mathematical Monthly, 86 (1979), 356-372.
- Dudley R.M., “Some inequalities for continued fractions.” Mathematics of Computation, 49 (180), pp. 585-593 (Oct 1987).
- 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).
- Ferraro G, Panza, M., “Developing into series and returning from series: A note on the foundations of eighteenth-century analysis.” Historia Mathematica, 30 (1), pp. 17-46 (Feb 2003).
- Fraser C.G., “The calculus as algebraic analysis - some observations on mathematical-analysis in the 18th-century.” Archive for History of Exact Sciences, 39 (4), pp. 317-335 (1989).
- Golland L.A, Golland RW., “Euler troublesome series - an early example of the use of trigonometric series.” Historia Mathematica, 20 (1), pp. 54-67, (Feb 1993).
- Pieper H., “On Euler contributions to the 4-squares theorem.” Historia Mathematica, 20 (1), pp. 12-18 (Feb 1993).
Return to the Euler Archive