E189 -- De serierum determinatione seu nova methodus inveniendi terminos generales serierum

(On the determination of series, or a new method for finding the general terms of series)


This paper attempts to determine f(x), given f(1), f(2), f(3), etc. Euler starts with an example: f(n) = n and tries f(x) = x + Sn=1bn sin(npx). Note that f(x) - x must be periodic and that f(x+1) = f(x). Then he gets f(x+1) by Taylor series at f(x) and gets a differential equation of infinite order. A few paragraphs later, we have a perfect Fourier series.

According to the records, it was presented to the St. Petersburg Academy on September 21, 1750.

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