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The classical “no-9” Kempner series is
One can represent exactly the Kempner series via alternating (or positive) series having geometric convergence, making the computation of their values with tens of thousands of digits amenable to personal computers. This involves coefficients related to the moments of certain measures. See Moments in the exact summation of the curious series of Kempner type (The American Mathematical Monthly 132:10 (2025), 995-1006). The last preprint version is at arXiv:2402.08525.
The present repository is initially to document implementations (back in prehistoric times in 2012) of the formulas which were published in the above reference. The first installment of this repository contains only, besides this short description, the two Maple files I attached in 2024 to the original preprint version posted on arXiv.
Important
The two Maple files kempner_base10_all.mpl and kempner_2000digits.mpl are both almost identical up to translation into English with some earlier source I wrote in January 2012.
The file kempner_2000digits.mpl is now completely superseded by the SageMath code I wrote in February 2024 to accompany my next manuscript Measures for the summation of Irwin series. See the sibling project burnolmath/irwin.
As per kempner_base10_all.mpl, it handles only radix 10, but remains to this day the sole implementation of the formulas of my Kempner paper for more than one digit being excluded. I created this repository with the uncertain perspective to update it later with some new and more powerful numerical implementations.
I do not document how to use the Maple files with extensions
.mpl (but see comments therein) because I hold a Maple license
only on some old hardware I am not using anymore, so I couldn't
test now my own instructions. This is also the reason why I do
not attach the files with extension .mw. However if you are
familiar with Maple syntax, you will have not issue using the
.mpl files. Anyhow I am also attaching the outputs I obtained from
them in February 2024.
This repository is devoted to the formulas from:
- Moments in the exact summation of the curious series of Kempner type, The American Mathematical Monthly 132:10 (2025), 995-1006 (DOI, preprint version at arXiv:2402.08525).
The formulas are generalized to Irwin series in:
- Measures for the summation of Irwin series, Integers 26:A11 (2026), 20pp (DOI).
Refer to the sibling project burnolmath/irwin for details and additional bibliographical references.
The files in this repository are distributed under the CC-BY-SA 4.0 License. See LICENSE.