E447  Summatio progressionum
\begin{eqnarray}
\sin\phi^\lambda + \sin 2\phi^\lambda + \sin 3\phi^\lambda + \cdots + \sin n\phi^\lambda; \\
\cos\phi^\lambda + \cos 2\phi^\lambda + \cos 3\phi^\lambda + \cdots + \cos n\phi^\lambda
\end{eqnarray}
(The summation of the progressions
\begin{eqnarray}
\sin\phi^\lambda + \sin 2\phi^\lambda + \sin 3\phi^\lambda + \cdots + \sin n\phi^\lambda; \\
\cos\phi^\lambda + \cos 2\phi^\lambda + \cos 3\phi^\lambda + \cdots + \cos n\phi^\lambda)
\end{eqnarray}
Summary:
This paper starts like E246 with the substitutions of u and v into an infinite series. They diverge if λ > 1. As Enlin Pan points out in his commentary, though, there is more going on here. Before the paper is over, Euler generalizes the notion of limit, and comes quite close to developing basic ideas of Fourier series and Cesaro sums. There is much of interest here.
According
to the records, it was presented to the St. Petersburg Academy on November 22, 1773.
Publication:

Originally published in Novi Commentarii academiae scientiarum Petropolitanae 18, 1774, pp. 2436

Opera Omnia: Series 1, Volume 15, pp. 168  184
Documents Available:
 Original publication: E447
 Enlin Pan has completed a translation of this paper, along with a short commentary: translation of E447.
 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 E447 include:
 Cooke R., “Uniqueness of trigonometric series and descriptive settheory, 18701985.” Archive for History of Exact Sciences, 45 (4), pp. 281334 (1993).
 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).
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