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Nilakantha's accelerated series for π

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Brink D.

Acta Arithmetica

2015

tom 171

nr 4

293-308

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.

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1 Acta Arithmetica

1 Brink D.

1 2015

Wyniki wyszukiwania

Nilakantha's accelerated series for π