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On infinite series representations of real numbers

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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. <> © Foundation Compositio Mathematica, 1973, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http:// implique l’accord avec les conditions générales d’utilisation ( 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 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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