Nilakantha's accelerated series for $\pi $
Tom 171 / 2015
Acta Arithmetica 171 (2015), 293-308
MSC: Primary 65B10; Secondary 40A25.
DOI: 10.4064/aa171-4-1
Streszczenie
We show how the idea behind a formula for $\pi $ 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 \[\pi = \sum _{n=0}^\infty \frac {(5n+3) n! (2n)!}{2^{n-1} (3n+2)!}\] with convergence as $13.5^{-n}$, in much the same way as the Euler transformation gives \[ \pi = \sum _{n=0}^\infty \frac {2^{n+1} n! n!}{(2n+1)!} \] with convergence as $2^{-n}$. Similar transformations lead to other accelerated series for $\pi $, including three ‶BBP-like″ formulas, all of which are collected in the Appendix. Optimal convergence is achieved using Chebyshev polynomials.
