Representations of Real Numbers by Infinite Series

Transcendence of a fast converging series of rational numbers

“…The oldest result on the irrationality of fast converging series seems to be due to Sylvester [24], who proved in 1880 For a proof of Theorem 2.1, see [24], [22,Chapter 4], [11,Chapter 1], [5,Exercice 2.9]. As a matter of fact, it is very easy to verify that α in (2.1) is rational when (2.2) is satisfied ; indeed…”

Metric properties of alternating Oppenheim expansions

“…These notes were used by Galambos [1] where the foundations of the metric theory of (2)-(4) are laid down. Further development, with several new results, can be found in a monograph of Galambos [2]. The expansion (2)-(4) became known as Oppenheim expansion.…”

“…The algorithm (2) never terminates, and (3) with (4) is equivalent to (2). The representation (3) under (4) is unique.…”

Engel Series Expansions of Laurent Series and Hausdorff Dimensions

“…As a matter of fact, in all practical applications we replace arbitrary 'numbers' by their decimal expansions after a certain number of 'digits'. Recently Knopfmacher and Knopfmacher [8,9] introduced and studied some properties of various unique expansions of formal Laurent series over a field F, as the sums of reciprocals of polynomials, involving 'digits' a x , a 2 ,... lying in a polynomial ring F[z] over F. In particular, one of these expansions was constructed to be analogous to the so-called Engel expansion of a real number, discussed in Galambos [5]. A number 2 Jun Wu [2] of famous expansions including those of Euler and the Rogers-Ramanujan identities are, in fact, special cases of Engel expansions of formal Laurent series.…”