E130  De seriebus quibusdam considerationes
(Considerations on certain series)
Summary:
Euler factors a power series into an infinite product, applying it to f'/f, and he evaluates a number of series that look
like trigonometric substitutions. He also applies this to the differential equation zs'  s  sz + s^{2} = 0 by recursion. Then he takes a shot at
z(2k+1).
According to
the records, it was presented to the St. Petersburg Academy on October 22, 1739.
Publication:

Originally published in Commentarii academiae scientiarum Petropolitanae 12, 1750, pp. 5396

Opera Omnia: Series 1, Volume 14, pp. 407  462
Documents Available:
 Original publication: E130
 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 E130 include:
 Dutka J., “On the summation of some divergent series of Euler and the zeta functions.” Archive for History of Exact Sciences, 50 (2), pp. 187200 (1996).
 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).
 Kline M., “Euler and infinite series.” Mathematics Magazine, 56 (5), pp. 307314 (1983).
Return to the Euler Archive