Badea, C. The irrationality of certain infinite series. (English) Zbl 0629.10027 Glasg. Math. J. 29, 221-228 (1987). The article is designed to prove the irrationality of a certain class of series in the same way as results due to P. Erdős, E. G. Straus and more recently to J. Sandor and the author himself. The main theorem proves the irrationality of \(\sum^{\infty}_{n=1} b_n/a_n\) (provided that series converges) when \((a_n)\) and \((b_n)\) are sequences of positive integers such that \((a_{n+1}-1)b_n>(a^2_n-a_n)b_{n+1}\) (at least for \(n\) big enough). The author provides a lot of interesting corollaries. For example, if \((F_n)\) is the Fibonacci sequence and \((L_n)\) the Lucas sequence \((L_n=F_{n-1}+F_{n+1})\) then \[ \sum^{\infty}_{n=1} 1/F_{2^n+1}\quad\text{and}\quad \sum^{\infty}_{n=1} 1/L_{2^n}\] are irrational numbers. Reviewer: Alain Escassut Cited in 4 ReviewsCited in 16 Documents MSC: 11J72 Irrationality; linear independence over a field 11B39 Fibonacci and Lucas numbers and polynomials and generalizations Keywords:infinite series; sums of reciprocal values; irrationality; Fibonacci sequence; Lucas sequence PDF BibTeX XML Cite \textit{C. Badea}, Glasg. Math. J. 29, 221--228 (1987; Zbl 0629.10027) Full Text: DOI OpenURL Online Encyclopedia of Integer Sequences: Decimal expansion of Sum_{k>=0} 1/L(2^k), where L(k) is the k-th Lucas number (A000032). Decimal expansion of Sum_{k>=0} 1/F(2^k+1), where F(k) is the k-th Fibonacci number (A000045). References: [1] C&lin, Gaz. Mat. Ser. A 75 pp 161– (1970) [2] Brun, Arch, for Math, og Naturvidenskab (Kristiania) 31 pp 3– (1910) [3] Sándor, Studia Univ. Babeş-Bolyai Math. 29 pp 3– (1984) [4] Mahler, Bull. Austral. Math. Soc. 13 pp 389– (1975) [5] Hoggatt, Fibonacci Quart 14 pp 453– (1976) [6] Erdös, J. Math. Sci. 10 pp 1– (1975) [7] Gelbaum, Problems in analysis (1982) · Zbl 0494.00004 [8] Froda, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 35 pp 472– (1963) [9] Good, Fibonacci Quart 12 (1974) [10] Erdös, J. Indian Math. Soc. 27 pp 129– (1963) [11] Erdös, Old and new problems and results in combinatorial number theory (1980) [12] Guy, Unsolved problems in number theory (1981) · Zbl 0474.10001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.