@article {MATHEDUC.05519666,
author = {Schmelzer, Thomas and Baillie, Robert},
title = {Summing a curious, slowly convergent series.},
year = {2008},
journal = {American Mathematical Monthly},
volume = {115},
number = {6},
issn = {0002-9890},
pages = {525-540},
publisher = {Mathematical Association of America (MAA), Washington, DC},
abstract = {It is well known that unlike the harmonic series the series $\sum\nolimits^{(9)} 1/n$ over all reciprocals of integers that do not contain the digit~$9$ in their base~$10$ representation converges. This is likewise true of all series $\sum\nolimits^{(X)} 1/n$ in which $n$ is confined to the set of integers that do not contain the string $X$ in their base~$10$ representation. The purpose of the paper is to present an efficient algorithm for computing the value $\psi_X$ of such a series to an accuracy of 100~digits. (Actually, to compute $\psi_{42}$ was a challenge put forward by [{\it F.~Bornemann, D.~Laurie, S.~Wagon, J.~Waldvogel}, The SIAM 100-digit challenge. A study in high-accuracy numerical computing. (Philadelphia), PA: Society for Industrial and Applied Mathematics (SIAM). (2004; Zbl 1060.65002)]). The algorithm uses truncation and extrapolation methods and is linked to the Perron-Frobenius theorem on spectral properties of matrices with positive entries. The authors conjecture that for a sequence of $n$-digit strings $X_n$ without periodic patterns $\lim_n \psi_{X_n}/10^n = \log 10$ and prove a related conjecture for 1-periodic~$X_n$.},
reviewer = {Dirk Werner (Berlin)},
msc2010 = {I35xx},
identifier = {2013a.00693},
}