
05519666
j
2013a.00693
Schmelzer, Thomas
Baillie, Robert
Summing a curious, slowly convergent series.
Am. Math. Mon. 115, No. 6, 525540 (2008).
2008
Mathematical Association of America (MAA), Washington, DC
EN
I35
slowly convergent series
computation
algorithm
extrapolation
truncation
Zbl 1060.65002
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 100digit challenge. A study in highaccuracy 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 PerronFrobenius 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 1periodic~$X_n$.
Dirk Werner (Berlin)