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Summing a curious, slowly convergent series. (English)
Am. Math. Mon. 115, No. 6, 525-540 (2008).
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 $ψ_X$ of such a series to an accuracy of 100~digits. (Actually, to compute $ψ_{42}$ 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 $\lim_n ψ_{X_n}/10^n = \log 10$ and prove a related conjecture for 1-periodic~$X_n$.
Reviewer: Dirk Werner (Berlin)
Classification: I35