Badea, C. The irrationality of certain infinite products. (English) Zbl 0625.10027 Stud. Univ. Babeș-Bolyai, Math. 31, No. 3, 3-8 (1986). The author considers a convergent product \(\prod^{\infty}_{n=1}(1+b_ n/a_ n)\) with \(a_ n\), \(b_ n\) positive integers, and he gives a condition to have the product irrational. For example, given a positive integer k, \(\prod^{\infty}_{n=1}(1+k/a_ n)\) is irrational provided the inequality \(a_{n+1}>a^ 2_ n+(k-1)a_ n+1-k\) is true for n large enough. This result extends a theorem of G. Cantor (1932) and also a result due to I. Sándor. He also notices connections with a false assertion made by Froda on V. Brun’s irrationality criterion [Arch. f. Math. og Nat. 31, No. 3, 6 p. (1910; Zbl 1234.11092)]. Reviewer: A.Escassut MSC: 11J81 Transcendence (general theory) Keywords:infinite products; Cantor’s theorem; convergent product; irrationality Citations:Zbl 1234.11092 PDF BibTeX XML Cite \textit{C. Badea}, Stud. Univ. Babeș-Bolyai, Math. 31, No. 3, 3--8 (1986; Zbl 0625.10027) OpenURL