General observations on series whose terms proceed as the sines and cosines of multiples of angles

TL;DR

This paper explores series involving sine and cosine terms of multiples of angles, analyzing their properties and expansions, with historical insights from Euler's work on binomial expansions involving complex exponentials.

Contribution

It provides a detailed examination of series with sinusoidal terms and connects these to Euler's earlier work on binomial expansions with complex numbers.

Findings

Series convergence properties analyzed

Connections between sinusoidal series and binomial expansions established

Special cases for half-integer exponents studied

Abstract

Translated from the Latin original, "Observationes generales circa series, quarum termini secundum sinus vel cosinus angulorum multiplorum progrediuntur" (1777). E655 in the Enestrom index. Euler looks at the binomial expansion $(1+x)^n$ for $x=\cos \phi+i\sin\phi$. He does $n=$ positive and negative integers, and $n={1/2}, -{1/2}$, and particular values of $\phi$.