On the irrationality of certain super-polynomially decaying series

2025/04/25 by Crmarić, Tonći, Kovač, Vjekoslav · 2 citations
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.18712

Abstract

We give a negative answer to a question by Paul Erdős and Ronald Graham on whether the series ∑_n=1^∞ (1)/((n+1)(n+2)⋯(n+f(n))) has an irrational sum whenever (f(n))_n=1^∞ is a sequence of positive integers converging to infinity. To achieve this, we generalize a classical observation of Sōichi Kakeya on the set of all subsums of a convergent positive series. We also discuss why the same problem is likely difficult when (f(n))_n=1^∞ is additionally assumed to be increasing.

Citations

• On several irrationality problems for Ahmes series
• On the set of points represented by harmonic subseries

Cited by

• Lacunary sequences whose reciprocal sums represent all rational numbers in an interval
• Irrationality of rapidly converging series: A problem of Erdős and Graham

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