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Shooting for π: the bowstring lemma

Published online by Cambridge University Press:  01 August 2016

Michael S. Longuet-Higgins
Michael S. Longuet-Higgins*
Affiliation:
Institute for Nonlinear Science, University of California, San Diego, La Jolla, California 92093-0402
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Extract

Some remarkable new series for π, in which the convergence is exponentially rapid, have been discovered by Bailey et al. [1, 2] using machine subroutines; see also Wagon [3]. An example is equation (17) below. Although the authors also provide algebraic proofs, these seem at first sight to be very special. The purpose of this note is to provide a more general demonstration and to deduce other instances of such series. It will be shown that all the results follow naturally from one or other of two simple lemmas.

Type
Research Article
Information
The Mathematical Gazette , Volume 84 , Issue 500 , July 2000 , pp. 216 - 222
Copyright
Copyright © The Mathematical Association 2000

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References

1. Bailey, D., Borwein, P. and Plouffe, S., On the rapid computation of various polylogarithmic constants, Math. Computation 66 (1997) pp. 915–925.CrossRefGoogle Scholar
2. Bailey, D. H., Borwein, J. M., Borwein, P. B. and Plouffe, S., The quest for Pi, Math. Intelligencer 19 (1) (1997) pp. 50–57.Google Scholar
3. Wagon, S., Book review, Math. Intelligencer 19 (3) (1997) pp. 59–67.CrossRefGoogle Scholar
4. Adamchik, V. and Wagon, S., A simple formula for π, Amer. Math. Monthly 104 (9) (1997) pp. 852–855.Google Scholar
5. Berggren, J., Borwein, J. and Borwein, P., Pi: A source book, Springer-Verlag, Berlin (1997), pp. 716.Google Scholar