2008/07/22 by Bakir Farhi, Farhi, Bakir
Mathematics · #40A05 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT) #math.NT #msc:40A05
paper · pdf · doi:10.48550/arxiv.0807.3518
5 pages, to appear in (The) American Mathematical Monthly
arxiv created 2008/07/22 · openalex publication_date 2008/07/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well known since A. J. Kempner's work that the series of the reciprocals of the positive integers whose the decimal representation does not contain any digit 9, is convergent. This result was extended by F. Irwin and others to deal with the series of the reciprocals of the positive integers whose the decimal representation contains only a limited quantity of each digit of a given nonempty set of digits. Actually, such series are known to be all convergent. Here, letting S(r) (r ∈ ℕ) denote the series of the reciprocal of the positive integers whose the decimal representation contains the digit 9 exactly r times, the impressive obtained result is that S(r) tends to 10 log10 as r tends to infinity!