Editing Pi (section)

=== Rapidly convergent series ===
[[File:Srinivasa Ramanujan - OPC - 2 (cleaned).jpg|thumb|upright=0.8|alt=Photo portrait of a man| [[Srinivasa Ramanujan]], working in isolation in India, produced many innovative series for computing {{pi}}.]]
Modern {{pi}} 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.{{r|Background}} The fast iterative algorithms were anticipated in 1914, when Indian mathematician [[Srinivasa Ramanujan]] published dozens of innovative new formulae for {{pi}}, remarkable for their elegance, mathematical depth and rapid convergence.{{sfn|Arndt|Haenel|2006|pp=103–104}} One of his formulae, based on [[modular equation]]s, is

<math display=block>
\frac{1}{\pi} = \frac{2 \sqrt 2}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{k!^4\left(396^{4k}\right)}.
</math>

This series converges much more rapidly than most arctan series, including Machin's formula.{{sfn|Arndt|Haenel|2006|p=104}}[[Bill Gosper]] was the first to use it for advances in the calculation of {{pi}}, setting a record of 17 million digits in 1985.{{sfn|Arndt|Haenel|2006|pp=104, 206}} Ramanujan's formulae anticipated the modern algorithms developed by the Borwein brothers ([[Jonathan Borwein|Jonathan]] and [[Peter Borwein|Peter]]) and the [[Chudnovsky brothers]].{{sfn|Arndt|Haenel|2006|pp=110–111}} The [[Chudnovsky algorithm|Chudnovsky formula]] developed in 1987 is

<math display=block>
\frac{1}{\pi} = \frac{\sqrt{10005}}{4270934400} \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!\,k!^3 (-640320)^{3k}}.
</math>

It produces about 14 digits of {{pi}} per term{{sfn|Eymard|Lafon|2004|p=254}} and has been used for several record-setting {{pi}} calculations, including the first to surpass 1 billion (10<sup>9</sup>) digits in 1989 by the Chudnovsky brothers, 10 trillion (10<sup>13</sup>) digits in 2011 by Alexander Yee and Shigeru Kondo,<ref name="NW">{{cite book |last1=Bailey |first1=David H. |author1-link=David H. Bailey (mathematician) |last2=Borwein |first2=Jonathan M. |author2-link=Jonathan Borwein |contribution=15.2 Computational records |contribution-url=https://books.google.com/books?id=K26zDAAAQBAJ&pg=PA469 |doi=10.1007/978-3-319-32377-0 |page=469 |publisher=Springer International Publishing |title=Pi: The Next Generation, A Sourcebook on the Recent History of Pi and Its Computation |year=2016 |isbn=978-3-319-32375-6}}</ref> and 100 trillion digits by [[Emma Haruka Iwao]] in 2022.<ref>{{Cite magazine |url=https://thenewstack.io/how-googles-emma-haruka-iwao-helped-set-a-new-record-for-pi/ |title=How Google's Emma Haruka Iwao Helped Set a New Record for Pi |date=11 June 2022 |magazine=The New Stack |first=David |last=Cassel}}</ref><ref>{{Cite web |last=Haruka Iwao |first=Emma |author-link=Emma Haruka Iwao |url=https://cloud.google.com/blog/products/compute/calculating-31-4-trillion-digits-of-archimedes-constant-on-google-cloud |title=Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedes' constant on Google Cloud |website=[[Google Cloud Platform]] |date=14 March 2019 |access-date=12 April 2019 |archive-url=https://web.archive.org/web/20191019023120/https://cloud.google.com/blog/products/compute/calculating-31-4-trillion-digits-of-archimedes-constant-on-google-cloud |archive-date=19 October 2019 |url-status=live}}</ref>  For similar formulae, see also the [[Ramanujan–Sato series]].

In 2006, mathematician [[Simon Plouffe]] used the PSLQ [[integer relation algorithm]]<ref>PSLQ means Partial Sum of Least Squares.</ref> to generate several new formulae for {{pi}}, conforming to the following template:

<math display=block>
\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),
</math>

where {{math|''q''}} is {{math|[[Gelfond's constant|''e''<sup>''&pi;''</sup>]]}} (Gelfond's constant), {{math|''k''}} is an [[odd number]], and {{math|''a'', ''b'', ''c''}} are certain rational numbers that Plouffe computed.<ref>{{cite web |first=Simon |last=Plouffe |author-link=Simon Plouffe |title=Identities inspired by Ramanujan's Notebooks (part 2) |date=April 2006 |url=<!-- http://www.lacim.uqam.ca/~plouffe/inspired2.pdf -->http://plouffe.fr/simon/inspired2.pdf |access-date=10 April 2009 |url-status=live |archive-url=https://web.archive.org/web/20120114101641/http://www.plouffe.fr/simon/inspired2.pdf |archive-date=14 January 2012}}</ref>