Editing Pi (section)

=== Infinite series ===
{{comparison_pi_infinite_series.svg}}
The calculation of {{pi}} was revolutionized by the development of [[infinite series]] techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite [[sequence (mathematics)|sequence]]. Infinite series allowed mathematicians to compute {{pi}} with much greater precision than [[Archimedes]] and others who used geometrical techniques.{{sfn|Arndt|Haenel|2006|pp=185–191}} Although infinite series were exploited for {{pi}} most notably by European mathematicians such as [[James Gregory (mathematician)|James Gregory]] and [[Gottfried Wilhelm Leibniz]], the approach also appeared in the [[Kerala school of astronomy and mathematics|Kerala school]] sometime in the 14th or 15th century.{{r|Roypp}}{{sfn|Arndt|Haenel|2006|pp=185–186}} Around 1500, an infinite series that could be used to compute {{pi}}, written in the form of [[Sanskrit]] verse, was presented in ''[[Tantrasamgraha]]'' by [[Nilakantha Somayaji]].<ref name="Roypp">{{cite journal |last=Roy |first=Ranjan |author-link=Ranjan Roy |year=1990 |title=The Discovery of the Series Formula for {{mvar|π}} by Leibniz, Gregory and Nilakantha |journal=Mathematics Magazine |volume=63 |number=5 |pages=291–306 |url=https://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf |doi=10.1080/0025570X.1990.11977541 |access-date=21 February 2023 |archive-date=14 March 2023 |archive-url=https://web.archive.org/web/20230314224252/https://www.maa.org/sites/default/files/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf |url-status=dead}}</ref> The series are presented without proof, but proofs are presented in the later work ''[[Yuktibhāṣā]]'', published around 1530. Several infinite series are described, including series for sine (which Nilakantha attributes to [[Madhava of Sangamagrama]]), cosine, and arctangent which are now sometimes referred to as [[Madhava series]]. The series for arctangent is sometimes called [[Gregory's series]] or the Gregory–Leibniz series.{{r|Roypp}} Madhava used infinite series to estimate {{pi}} to 11 digits around 1400.<ref>{{cite book |last=Joseph |first=George Gheverghese |author-link=George Gheverghese Joseph |title=The Crest of the Peacock: Non-European Roots of Mathematics |publisher=Princeton University Press |year=1991 |isbn=978-0-691-13526-7 |url=https://books.google.com/books?id=c-xT0KNJp0cC&pg=PA264 |page=264}}<!-- This ISBN is for the third edition from 2011! --></ref>{{sfn|Andrews|Askey|Roy|1999|p=59}}<ref>{{Cite journal |first=R. C. |last=Gupta |author-link=Radha Charan Gupta |title=On the remainder term in the Madhava–Leibniz's series |journal=Ganita Bharati |volume=14 |issue=1–4 |year=1992 |pages=68–71}}</ref>

In 1593, [[François Viète]] published what is now known as [[Viète's formula]], an [[infinite product]] (rather than an [[infinite sum]], which is more typically used in {{pi}} calculations):<ref>{{harvnb|Arndt|Haenel|2006|p=187}}. {{pb}} {{cite book |url=https://books.google.com/books?id=7_BCAAAAcAAJ |title=Variorum de rebus mathematicis responsorum |volume=VIII |first=Franciscus |last=Vieta |year=1593}} {{pb}} {{OEIS2C|id=A060294}}</ref>

<math display=block> \frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots</math>

In 1655, [[John Wallis]] published what is now known as [[Wallis product]], also an infinite product:{{sfn|Arndt|Haenel|2006|p=187}}

<math display=block>
\frac{\pi}{2} = \Big(\frac{2}{1} \cdot \frac{2}{3}\Big) \cdot \Big(\frac{4}{3} \cdot \frac{4}{5}\Big) \cdot \Big(\frac{6}{5} \cdot \frac{6}{7}\Big) \cdot \Big(\frac{8}{7} \cdot \frac{8}{9}\Big) \cdots
</math>

[[File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|upright=0.8|alt=A formal portrait of a man, with long hair|[[Isaac Newton]]
used [[infinite series]] to compute {{pi}} to 15 digits, later writing "I am ashamed to tell you to how many figures I carried these computations".{{r|Newton}}]]
In the 1660s, the English scientist [[Isaac Newton]] and German mathematician [[Gottfried Wilhelm Leibniz]] discovered [[calculus]], which led to the development of many infinite series for approximating {{pi}}. Newton himself used an arcsine series to compute a 15-digit approximation of {{pi}} in 1665 or 1666, writing, "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."<ref name="Newton">{{harvnb|Arndt|Haenel|2006|p=188}}. Newton quoted by Arndt.</ref>

In 1671, [[James Gregory (mathematician)|James Gregory]], and independently, Leibniz in 1673, discovered the [[Taylor series]] expansion for [[arctangent]]:{{r|Roypp}}<ref>{{cite journal |last=Horvath |first=Miklos |title=On the Leibnizian quadrature of the circle. |journal=Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica) |volume=4 |year=1983 |pages=75–83 |url=http://ac.inf.elte.hu/Vol_004_1983/075.pdf}}</ref>{{sfn|Eymard|Lafon|2004|pp=53–54}}

<math display=block>
\arctan z = z - \frac {z^3} {3} +\frac {z^5} {5} -\frac {z^7} {7} +\cdots
</math>

This series, sometimes called the [[Gregory's series|Gregory–Leibniz series]], equals <math display="inline">\frac{\pi}{4}</math> when evaluated with <math>z=1</math>.{{sfn|Eymard|Lafon|2004|pp=53–54}} But for <math>z=1</math>, [[Leibniz formula for π#Convergence|it converges impractically slowly]] (that is, approaches the answer very gradually), taking about ten times as many terms to calculate each additional digit.<ref>{{cite journal |last=Cooker |first=M. J. |year=2011 |title=Fast formulas for slowly convergent alternating series |journal=Mathematical Gazette |volume=95 |number=533 |pages=218–226 |doi=10.1017/S0025557200002928 |s2cid=123392772 |url=https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F7C083868DEB95FE049CD44163367592/S0025557200002928a.pdf/div-class-title-fast-formulas-for-slowly-convergent-alternating-series-div.pdf |access-date=23 February 2023 |archive-date=4 May 2019 |archive-url=https://web.archive.org/web/20190504091131/https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F7C083868DEB95FE049CD44163367592/S0025557200002928a.pdf/div-class-title-fast-formulas-for-slowly-convergent-alternating-series-div.pdf |url-status=deviated <!-- now paywalled -->}}</ref>

In 1699, English mathematician [[Abraham Sharp]] used the Gregory–Leibniz series for <math display="inline">z=\frac{1}{\sqrt{3}}</math> to compute {{pi}} to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.{{sfn|Arndt|Haenel|2006|p=189}}

In 1706, [[John Machin]] used the Gregory–Leibniz series to produce an algorithm that converged much faster:<ref name=jones>{{cite book |last=Jones |first=William |author-link=William Jones (mathematician) |year=1706 |title=Synopsis Palmariorum Matheseos |place=London |publisher=J. Wale |url=https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n283/ |pages=[https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n261/ 243], [https://archive.org/details/SynopsisPalmariorumMatheseosOrANewIntroductionToTheMathematics/page/n283/ 263] |quote-page=263 |quote=There are various other ways of finding the ''Lengths'', or ''Areas'' of particular ''Curve Lines'' or ''Planes'', which may very much facilitate the Practice; as for instance, in the ''Circle'', the Diameter is to Circumference as 1 to {{br}}<math>
\overline{\tfrac{16}5 - \tfrac4{239}}
- \tfrac13\overline{\tfrac{16}{5^3} - \tfrac4{239^3}}
+ \tfrac15\overline{\tfrac{16}{5^5} - \tfrac4{239^5}}
-,\, \&c. =</math>{{br}}{{math|1=3.14159, &''c.'' = ''π''}}. This ''Series'' (among others for the same purpose, and drawn from the same Principle) I receiv'd from the Excellent Analyst, and my much Esteem'd Friend Mr. ''[[John Machin]]''; and by means thereof, ''[[Ludolph van Ceulen|Van Ceulen]]''{{'}}s Number, or that in Art. 64.38. may be Examin'd with all desireable Ease and Dispatch.
}} {{pb}} Reprinted in {{cite book |last=Smith |first=David Eugene |author-link=David Eugene Smith |year=1929 |title=A Source Book in Mathematics |publisher=McGraw–Hill |chapter=William Jones: The First Use of {{mvar|π}} for the Circle Ratio |chapter-url=https://archive.org/details/sourcebookinmath1929smit/page/346/ |pages=346–347}}</ref><ref name=tweddle>{{cite journal |last=Tweddle |first=Ian |year=1991 |title=John Machin and Robert Simson on Inverse-tangent Series for {{mvar|π}} |journal=Archive for History of Exact Sciences |volume=42 |number=1 |pages=1–14 |doi=10.1007/BF00384331 |jstor=41133896 |s2cid=121087222}}</ref>{{sfn|Arndt|Haenel|2006|pp=192–193}}

<math display=block> \frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}.</math>

Machin reached 100 digits of {{pi}} with this formula.{{sfn|Arndt|Haenel|2006|pp=72–74}} Other mathematicians created variants, now known as [[Machin-like formula]]e, that were used to set several successive records for calculating digits of {{pi}}.<ref>{{cite journal |last=Lehmer |first=D. H. |author-link=D. H. Lehmer |year=1938 |title=On Arccotangent Relations for π |journal=American Mathematical Monthly |volume=45 |number=10 |pages=657–664 Published by: Mathematical Association of America |jstor=2302434 |doi=10.1080/00029890.1938.11990873 |url=https://www.maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_7.pdf |access-date=21 February 2023 |archive-date=7 March 2023 |archive-url=https://web.archive.org/web/20230307164817/https://www.maa.org/sites/default/files/pdf/pubs/amm_supplements/Monthly_Reference_7.pdf |url-status=dead}}</ref>{{sfn|Arndt|Haenel|2006|pp=72–74}}

Isaac Newton [[series acceleration|accelerated the convergence]] of the Gregory–Leibniz series in 1684 (in an unpublished work; others independently discovered the result):<ref>{{cite book |last=Roy |first=Ranjan |author-link=Ranjan Roy |year=2021 |orig-year=1st ed. 2011 |title=Series and Products in the Development of Mathematics |edition=2 |volume=1 |publisher=Cambridge University Press |pages=215–216, 219–220}} {{pb}} {{cite book |last=Newton |first=Isaac |author-link=Isaac Newton |year=1971 |editor-last=Whiteside |editor-first=Derek Thomas |editor-link=Tom Whiteside |title=The Mathematical Papers of Isaac Newton |volume=4, 1674–1684 |publisher=Cambridge University Press |chapter=De computo serierum |trans-chapter=On the computation of series |at="De transmutatione serierum" [On the transformation of series] §{{nbsp}}3.2.2 {{pgs|604–615}} |chapter-url=https://archive.org/details/mathematicalpape0004newt/page/604/mode/2up |chapter-url-access=limited}}</ref>

<math display=block>
\arctan x
= \frac{x}{1 + x^2} + \frac23\frac{x^3}{(1 + x^2)^2}
+ \frac{2\cdot 4}{3 \cdot 5}\frac{x^5}{(1 + x^2)^3} + \cdots
</math>

[[Leonhard Euler]] popularized this series in his 1755 differential calculus textbook, and later used it with Machin-like formulae, including <math display=inline>\tfrac\pi4 = 5\arctan\tfrac17 + 2\arctan\tfrac{3}{79},</math> with which he computed 20 digits of {{pi}} in one hour.<ref>{{cite web |last=Sandifer |first=Ed |year=2009 |title=Estimating π |website=How Euler Did It |url=http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf}} Reprinted in {{cite book |last=Sandifer |first=Ed |display-authors=0 |year=2014 |title=How Euler Did Even More |pages=109–118 |publisher=Mathematical Association of America}} {{pb}} {{cite book |last=Euler |first=Leonhard |author-link=Leonhard Euler |year=1755 |title=[[Institutiones calculi differentialis|Institutiones Calculi Differentialis]] |chapter=§&nbsp;2.2.30 |page=318 |publisher=Academiae Imperialis Scientiarium Petropolitanae |language=la |chapter-url=https://archive.org/details/institutiones-calculi-differentialis-cum-eius-vsu-in-analysi-finitorum-ac-doctri/page/318 |id=[https://scholarlycommons.pacific.edu/euler-works/212/ E 212]}} {{pb}} {{cite journal |last=Euler |first=Leonhard |author-link=Leonhard Euler |year=1798 |orig-year=written 1779 |title=Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae |journal=Nova Acta Academiae Scientiarum Petropolitinae |volume=11 |pages=133–149, 167–168 |url=https://archive.org/details/novaactaacademia11petr/page/133 |id=[https://scholarlycommons.pacific.edu/euler-works/705/ E 705]}} {{pb}} {{cite journal |first=Chien-Lih |last=Hwang |year=2004 |title=88.38 Some Observations on the Method of Arctangents for the Calculation of {{mvar|π}} |journal=Mathematical Gazette |volume=88 |number=512 |pages=270–278 |doi=10.1017/S0025557200175060|jstor=3620848 |s2cid=123532808}} {{pb}} {{cite journal |first=Chien-Lih |last=Hwang |year=2005 |title=89.67 An elementary derivation of Euler's series for the arctangent function |journal=Mathematical Gazette |volume=89 |number=516 |pages=469–470 |doi=10.1017/S0025557200178404|jstor=3621947 |s2cid=123395287}}
</ref>

Machin-like formulae remained the best-known method for calculating {{pi}} well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson&nbsp;– the best approximation achieved without the aid of a calculating device.{{sfn|Arndt|Haenel|2006|pp=192–196, 205}}

In 1844, a record was set by [[Zacharias Dase]], who employed a Machin-like formula to calculate 200 decimals of {{pi}} in his head at the behest of German mathematician [[Carl Friedrich Gauss]].{{sfn|Arndt|Haenel|2006|pp=194–196}}

In 1853, British mathematician [[William Shanks]] calculated {{pi}} to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well.<ref name="hayes-2014">{{cite magazine |last=Hayes |first=Brian |author-link=Brian Hayes (scientist) |url=https://www.americanscientist.org/article/pencil-paper-and-pi |title=Pencil, Paper, and Pi |volume=102 |issue=5 |page=342 |magazine=[[American Scientist]] |date=September 2014 |access-date=22 January 2022 |doi=10.1511/2014.110.342}}</ref>

==== Rate of convergence ====
Some infinite series for {{pi}} [[convergent series|converge]] faster than others. Given the choice of two infinite series for {{pi}}, mathematicians will generally use the one that converges more rapidly because faster convergence reduces the amount of computation needed to calculate {{pi}} to any given accuracy.<ref name="Aconverge">{{cite journal |last1=Borwein |first1=J. M. |last2=Borwein |first2=P. B. |title=Ramanujan and Pi |year=1988 |journal=Scientific American |volume=256 |issue=2 |pages=112–117 |bibcode=1988SciAm.258b.112B |doi=10.1038/scientificamerican0288-112}}{{br}} {{harvnb|Arndt|Haenel|2006|pp=15–17, 70–72, 104, 156, 192–197, 201–202}}.</ref> A simple infinite series for {{pi}} is the [[Leibniz formula for π|Gregory–Leibniz series]]:{{sfn|Arndt|Haenel|2006|pp=69–72}}

<math display=block>
\pi = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \frac{4}{13} - \cdots
</math>

As individual terms of this infinite series are added to the sum, the total gradually gets closer to {{pi}}, and&nbsp;– with a sufficient number of terms&nbsp;– can get as close to {{pi}} as desired. It converges quite slowly, though&nbsp;– after 500,000 terms, it produces only five correct decimal digits of {{pi}}.<ref>{{cite journal |last1=Borwein |first1=J. M. |last2=Borwein |first2=P. B. |last3=Dilcher |first3=K. |year=1989 |title=Pi, Euler Numbers, and Asymptotic Expansions |journal=American Mathematical Monthly |volume=96 |issue=8 |pages=681–687 |doi=10.2307/2324715 |jstor=2324715 |hdl=1959.13/1043679 |hdl-access=free}}</ref>

An infinite series for {{pi}} (published by Nilakantha in the 15th century) that converges more rapidly than the Gregory–Leibniz series is:{{sfn|Arndt|Haenel|2006|loc = Formula 16.10, p. 223}}<ref>{{cite book |last=Wells |first=David |page=35 |title=The Penguin Dictionary of Curious and Interesting Numbers |edition=revised |publisher=Penguin |year=1997 |isbn=978-0-14-026149-3}}</ref>

<math display=block>
\pi = 3 + \frac{4}{2\times3\times4} - \frac{4}{4\times5\times6} + \frac{4}{6\times7\times8} - \frac{4}{8\times9\times10} + \cdots
</math>

The following table compares the convergence rates of these two series:

{|class="wikitable" style="text-align: center; margin: auto;"
|-
! Infinite series for {{pi}} !! After 1st term !! After 2nd term !! After 3rd term !! After 4th term !! After 5th term !! Converges to:
|-
| <math>\pi = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \frac{4}{11} + \frac{4}{13} + \cdots</math>
||4.0000||2.6666 ... ||3.4666 ... ||2.8952 ... ||3.3396 ... ||rowspan=2| {{pi}} = 3.1415 ...
|-
| <math>\pi = {{3}} + \frac{{4}}{2\times3\times4} - \frac{{4}}{4\times5\times6} + \frac{{4}}{6\times7\times8} - \cdots </math>
||3.0000||3.1666 ... ||3.1333 ... ||3.1452 ... ||3.1396 ...
|}

After five terms, the sum of the Gregory–Leibniz series is within 0.2 of the correct value of {{pi}}, whereas the sum of Nilakantha's series is within 0.002 of the correct value. Nilakantha's series converges faster and is more useful for computing digits of {{pi}}. Series that converge even faster include [[Machin-like formula|Machin's series]] and [[Chudnovsky algorithm|Chudnovsky's series]], the latter producing 14 correct decimal digits per term.{{r|Aconverge}}