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


11J72 Irrationality; linear independence over a field
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
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