# On infinite series representations of real numbers

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## Abstract

On infinite series representations of real numbers COMPOSITIO MATHEMATICA JÁNOSGALAMBOS On infinite series representations of real numbers Compositio Mathematica, tome 27, no 2 (1973), p. 197-204. <http://www.numdam.org/item?id=CM_1973__27_2_197_0> © Foundation Compositio Mathematica, 1973, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la pré- sente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 197 ON INFINITE SERIES REPRESENTATIONS OF REAL NUMBERS János Galambos COMPOSITIO MATHEMATICA, Vol. 27, Fasc. 2, 1973, pag. 197-204 Noordhoff International Publishing Printed in the Netherlands 1. Summary The major objective of the present paper is to generalize some of the results of Vervaat [12] and of the present author [1] and [6] in metric number theory by considering an algorithm which includes those investi- gated in the above papers. Though hints have been given for this more general expansion in the literature, metric results achieved their most gen- eral formulations in the quoted papers. Some of the results are new for the special cases of [1 ], [6] and [12], or even for the classical expansions of Engel, Sylvester and Cantor. 2. The algorithm Let xj (n) &#x3E; 0, j = 1, 2, ... be a sequence of strictly decreasing func- tions of natural numbers n and such that, for each j, 03B1j(1) = 1 and 03B1j(n)~ 0 as n - + oo . Let 03B3j(n) be another sequence of positive func- tions of n on which some further assumptions will be imposed in the sequel. Let 0 x 1 be an arbitrary real number and define the integers dj = dj(x) and the real numbers Xj by the algo

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