id: 05519666
dt: j
an: 2013a.00693
au: Schmelzer, Thomas; Baillie, Robert
ti: Summing a curious, slowly convergent series.
so: Am. Math. Mon. 115, No. 6, 525-540 (2008).
py: 2008
pu: Mathematical Association of America (MAA), Washington, DC
la: EN
cc: I35
ut: slowly convergent series; computation; algorithm; extrapolation; truncation
ci: Zbl 1060.65002
li:
ab: 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~$X_n$.
rv: Dirk Werner (Berlin)