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Irrationality of certain infinite series. (English) Zbl 1194.26029

Authors’ abstract: A new direct proof for the irrationality of Euler’s number
\[ e=\sum_{k=0}^\infty 1/k! \]
is presented. Furthermore, formulas for the base \(b\) digits are given which, however, are not computably effective. Finally we generalize our method and give a simple criterium for some fast converging series representing irrational numbers.

MSC:

26D15 Inequalities for sums, series and integrals
33C20 Generalized hypergeometric series, \({}_pF_q\)
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References:

[1] K. Knopp. Theorie und Anwendung der unendlichen Reihen. Springer, fifth edition, 1964. · Zbl 0124.28302
[2] W. Koepf. Hypergeometic Summation. Vieweg, 1998.
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