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A short proof of the simple continued fraction expansion of \(e\). (English) Zbl 1145.11012
The author presents a streamlined version of Hermite’s proof for Euler’s simple continued fraction expansion \(e= [2,1,2,1,1, 4,1,1,6,1,\dots]\). This proof rests on three sequences of integrals which are related to the convergents of this continued fraction in a simple and surprising way which has, however, no apparent motivation. The author shows that Padé approximants to the function \(e^z\) lead naturally to these integrals, and he provides evidence that Hermite found this approach, as a byproduct of his proof of the transcendence of Euler’s number \(e\), in the course of the 1892 thesis of his student H. Padé.

MSC:
11A55 Continued fractions
11J70 Continued fractions and generalizations
01A55 History of mathematics in the 19th century
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