Nilakantha's accelerated series for π

David Brink. "Nilakantha's accelerated series for π." Acta Arithmetica 171.4 (2015): 293-308. <http://eudml.org/doc/286514>.

@article{DavidBrink2015,
abstract = {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 $π = ∑_\{n=0\}^\{∞\} ((5n+3)n!(2n)!)/(2^\{n-1\}(3n+2)!)$ with convergence as $13.5^\{-n\}$, in much the same way as the Euler transformation gives $π = ∑_\{n=0\}^\{∞\} (2^\{n+1\}n!n!)/(2n+1)!$ 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.},
author = {David Brink},
journal = {Acta Arithmetica},
keywords = {series for ; convergence acceleration; Nilakantha; Gregory-Leibniz series; Euler transformation},
language = {eng},
number = {4},
pages = {293-308},
title = {Nilakantha's accelerated series for π},
url = {http://eudml.org/doc/286514},
volume = {171},
year = {2015},
}

TY - JOUR
AU - David Brink
TI - Nilakantha's accelerated series for π
JO - Acta Arithmetica
PY - 2015
VL - 171
IS - 4
SP - 293
EP - 308
AB - 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 $π = ∑_{n=0}^{∞} ((5n+3)n!(2n)!)/(2^{n-1}(3n+2)!)$ with convergence as $13.5^{-n}$, in much the same way as the Euler transformation gives $π = ∑_{n=0}^{∞} (2^{n+1}n!n!)/(2n+1)!$ 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.
LA - eng
KW - series for ; convergence acceleration; Nilakantha; Gregory-Leibniz series; Euler transformation
UR - http://eudml.org/doc/286514
ER -