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The irrationality of certain infinite series. (English) Zbl 0629.10027
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

MSC:
11J72 Irrationality; linear independence over a field
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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[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 · doi:10.1007/978-1-4615-7679-2
[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 · doi:10.1007/978-1-4757-1738-9
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