Nilakantha's accelerated series for π
David Brink
Acta Arithmetica, Tome 168 (2015), p. 293-308 / Harvested from The Polish Digital Mathematics Library
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We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula π=∑n=0∞((5n+3)n!(2n)!)/(2n-1(3n+2)!) with convergence as 13.5-n, in much the same way as the Euler transformation gives π=∑n=0∞(2n+1n!n!)/(2n+1)! with convergence as 2-n. Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in the Appendix. Optimal convergence is achieved using Chebyshev polynomials.

Publié le : 2015-01-01
EUDML-ID : urn:eudml:doc:286514 
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     title = {Nilakantha's accelerated series for $\pi$},
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     pages = {293-308},
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David Brink. Nilakantha's accelerated series for π. Acta Arithmetica, Tome 168 (2015) pp. 293-308. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-4-1/