section_path: Modern quest for more digits/Rapidly convergent series
section_name: Rapidly convergent series
paragraph_text: Rapidly convergent series
  
  Modern π calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive. The fast iterative algorithms were anticipated in 1914, when Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for π, remarkable for their elegance, mathematical depth and rapid convergence. One of his formulae, based on modular equations, is 1 π = 2 2 9801 ∑ k = 0 ∞ (4 k) ! (1103 + 26390 k) k ! 4 (396 4 k). {\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{k!^{4}\left(396^{4k}\right)}}.}
  
  This series converges much more rapidly than most arctan series, including Machin's formula. Bill Gosper was the first to use it for advances in the calculation of π, setting a record of 17 million digits in 1985. Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers (Jonathan and Peter) and the Chudnovsky brothers. The Chudnovsky formula developed in 1987 is 1 π = 10005 4270934400 ∑ k = 0 ∞ (6 k) ! (13591409 + 545140134 k) (3 k) ! k ! 3 (− 640320) 3 k. {\displaystyle {\frac {1}{\pi }}={\frac {\sqrt {10005}}{4270934400}}\sum _{k=0}^{\infty }{\frac {(6k)!(13591409+545140134k)}{(3k)!\,k!^{3}(-640320)^{3k}}}.}
  
  It produces about 14 digits of π per term and has been used for several record-setting π calculations, including the first to surpass 1 billion (10) digits in 1989 by the Chudnovsky brothers, 10 trillion (10) digits in 2011 by Alexander Yee and Shigeru Kondo, and 100 trillion digits by Emma Haruka Iwao in 2022. For similar formulae, see also the Ramanujan–Sato series.
  
  In 2006, mathematician Simon Plouffe used the PSLQ integer relation algorithm to generate several new formulae for π, conforming to the following template: π k = ∑ n = 1 ∞ 1 n k (a q n − 1 + b q 2 n − 1 + c q 4 n − 1), {\displaystyle \pi ^{k}=\sum _{n=1}^{\infty }{\frac {1}{n^{k}}}\left({\frac {a}{q^{n}-1}}+{\frac {b}{q^{2n}-1}}+{\frac {c}{q^{4n}-1}}\right),} where q is e (Gelfond's constant), k is an odd number, and a, b, c are certain rational numbers that Plouffe computed.