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<!doctype html> <html lang="en" class="calculusandanalysis discretemathematics foundationsofmathematics mathworldcontributors numbertheory"> <head> <title>Pi Formulas -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Pi Formulas" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="There are many formulas of pi of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. pi is intimately related to the properties of circles and spheres. For a circle of radius r, the circumference and area are given by C = 2pir (1) A = pir^2. (2) Similarly, for a sphere of radius r, the surface area and volume enclosed are S = 4pir^2 (3) V = 4/3pir^3. (4) An exact formula for pi in terms of the inverse tangents of..." /> <meta name="description" content="There are many formulas of pi of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. pi is intimately related to the properties of circles and spheres. For a circle of radius r, the circumference and area are given by C = 2pir (1) A = pir^2. (2) Similarly, for a sphere of radius r, the surface area and volume enclosed are S = 4pir^2 (3) V = 4/3pir^3. (4) An exact formula for pi in terms of the inverse tangents of..." /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2002-06-09" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2002-11-02" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2003-11-01" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2004-01-16" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2004-01-25" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2004-06-27" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2004-11-02" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2005-01-15" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2006-02-18" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2006-03-06" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2006-03-19" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2006-03-20" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2006-04-06" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2006-04-20" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2007-09-27" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2022-02-02" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Number Theory:Constants:Pi" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Discrete Mathematics:Experimental Mathematics" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Calculus and Analysis:Series:BBP Formulas" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Calculus and Analysis:Calculus:Integrals:Definite Integrals" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Foundations of Mathematics:Mathematical Problems:Unsolved Problems" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:MathWorld Contributors:Cloitre" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:MathWorld Contributors:Plouffe" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:MathWorld Contributors:Sondow" /> <meta name="DC.Rights" content="Copyright 1999-2024 Wolfram Research, Inc. See https://mathworld.wolfram.com/about/terms.html for a full terms of use statement." /> <meta name="DC.Format" scheme="IMT" content="text/html" /> <meta name="DC.Identifier" scheme="URI" content="https://mathworld.wolfram.com/PiFormulas.html" /> <meta name="DC.Language" scheme="RFC3066" content="en" /> <meta name="DC.Publisher" content="Wolfram Research, Inc." /> <meta name="DC.Relation.IsPartOf" scheme="URI" content="https://mathworld.wolfram.com/" /> <meta name="DC.Type" scheme="DCMIType" content="Text" /> <meta name="Last-Modified" content="2022-02-02" /> <meta property="og:image" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_PiFormulas.png"> <meta property="og:url" content="https://mathworld.wolfram.com/PiFormulas.html"> <meta property="og:type" content="website"> <meta property="og:title" content="Pi Formulas -- from Wolfram MathWorld"> <meta property="og:description" content="There are many formulas of pi of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. pi is intimately related to the properties of circles and spheres. For a circle of radius r, the circumference and area are given by C = 2pir (1) A = pir^2. (2) Similarly, for a sphere of radius r, the surface area and volume enclosed are S = 4pir^2 (3) V = 4/3pir^3. (4) An exact formula for pi in terms of the inverse tangents of..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Pi Formulas -- from Wolfram MathWorld"> <meta name="twitter:description" content="There are many formulas of pi of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. pi is intimately related to the properties of circles and spheres. For a circle of radius r, the circumference and area are given by C = 2pir (1) A = pir^2. (2) Similarly, for a sphere of radius r, the surface area and volume enclosed are S = 4pir^2 (3) V = 4/3pir^3. (4) An exact formula for pi in terms of the inverse tangents of..."> <meta name="twitter:image:src" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_PiFormulas.png"> <link rel="canonical" href="https://mathworld.wolfram.com/PiFormulas.html" /> <meta http-equiv="x-ua-compatible" content="ie=edge"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta charset="utf-8"> <script async src="/common/javascript/analytics.js"></script> <script async src="//www.wolframcdn.com/consent/cookie-consent.js"></script> <script async src="/common/javascript/wal/latest/walLoad.js"></script> <link rel="stylesheet" href="/css/styles.css"> <link rel="preload" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css" as="style" onload="this.onload=null;this.rel='stylesheet'"> <noscript><link rel="stylesheet" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css"></noscript> </head> <body id="topics"> <main id="entry"> <div class="wrapper"> <section 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History and Terminology </a> <a href="/topics/NumberTheory.html" id="sidebar-numbertheory"> Number Theory </a> <a href="/topics/ProbabilityandStatistics.html" id="sidebar-probabilityandstatistics"> Probability and Statistics </a> <a href="/topics/RecreationalMathematics.html" id="sidebar-recreationalmathematics"> Recreational Mathematics </a> <a href="/topics/Topology.html" id="sidebar-topology"> Topology </a> </nav> <nav class="secondary-nav"> <a href="/letters/"> Alphabetical Index </a> <a href="/whatsnew/"> New in MathWorld </a> </nav> </section> <section id="content">  <nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/NumberTheory.html">Number Theory</a> </li> <li> <a href="/topics/Constants.html">Constants</a> </li> <li> <a href="/topics/Pi.html">Pi</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/DiscreteMathematics.html">Discrete Mathematics</a> </li> <li> <a href="/topics/ExperimentalMathematics.html">Experimental Mathematics</a> </li> </ul><ul 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href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Plouffe.html">Plouffe</a> </li> </ul><ul class="breadcrumb"> <li> <a href="/topics/MathWorldContributors.html">MathWorld Contributors</a> </li> <li> <a href="/topics/Sondow.html">Sondow</a> </li> </ul><a class="show-more">More...</a><a class="display-n show-less">Less...</a></nav>   <h1>Pi Formulas</h1>  <hr class="margin-t-1-8 margin-b-3-4">  <div class="attachments text-align-r"> <a href="/notebooks/Constants/PiFormulas.nb" download="PiFormulas.nb"><img src="/images/entries/download-notebook-icon.png" width="26" height="27" alt="DOWNLOAD Mathematica Notebook" /><span>Download <span class="display-i display-n__600">Wolfram </span>Notebook</span></a> </div>  <div class="entry-content"> <p> There are many formulas of <img src="/images/equations/PiFormulas/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> of many types. Among others, these include series, products, geometric constructions, limits, special values, and <a href="/PiIterations.html">pi iterations</a>. </p> <p> <img src="/images/equations/PiFormulas/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> is intimately related to the properties of circles and spheres. For a circle of <a href="/Radius.html">radius</a> <img src="/images/equations/PiFormulas/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="7" height="21" alt="r" />, the circumference and area are given by </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline4.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="C" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline5.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline6.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="20" alt="2pir" /></td><td align="right" width="10"> <div id="eqn1" class="eqnum"> (1) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline7.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="11" height="20" alt="A" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline8.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline9.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="29" height="20" alt="pir^2." /></td><td align="right" width="10"> <div id="eqn2" class="eqnum"> (2) </div> </td></tr> </table> </div> <p> Similarly, for a sphere of <a href="/Radius.html">radius</a> <img src="/images/equations/PiFormulas/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="7" height="21" alt="r" />, the surface area and volume enclosed are </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline11.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="9" height="20" alt="S" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline12.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline13.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="37" height="20" alt="4pir^2" /></td><td align="right" width="10"> <div id="eqn3" class="eqnum"> (3) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline14.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="V" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline15.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline16.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="44" height="26" alt="4/3pir^3." /></td><td align="right" width="10"> <div id="eqn4" class="eqnum"> (4) </div> </td></tr> </table> </div> <p> An exact formula for <img src="/images/equations/PiFormulas/Inline17.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> in terms of the <a href="/InverseTangent.html">inverse tangents</a> of <a href="/UnitFraction.html">unit fractions</a> is <a href="/MachinsFormula.html">Machin's formula</a> </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation1.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="208" height="26" alt=" 1/4pi=4tan^(-1)(1/5)-tan^(-1)(1/(239)). " /></td><td align="right" width="3"> <div id="eqn5" class="eqnum"> (5) </div> </td></tr> </table> </div> <p> There are three other <a href="/Machin-LikeFormulas.html">Machin-like formulas</a>, as well as thousands of other similar formulas having more terms. </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="404.175" height="249.828" src="images/eps-svg/GregorySeries_1000.svg" class="" alt="GregorySeries" /> </div> <p> Gregory and Leibniz found </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline18.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="34" alt="pi/4" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline19.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline20.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="80" height="51" alt="sum_(k=1)^(infty)((-1)^(k+1))/(2k-1)" /></td><td align="right" width="10"> <div id="eqn6" class="eqnum"> (6) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline21.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline22.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline23.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="103" height="39" alt="1-1/3+1/5-..." /></td><td align="right" width="10"> <div id="eqn7" class="eqnum"> (7) </div> </td></tr> </table> </div> <p> (Wells 1986, p. 50), which is known as the <a href="/GregorySeries.html">Gregory series</a> and may be obtained by plugging <img src="/images/equations/PiFormulas/Inline24.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="x=1" /> into the <a href="/LeibnizSeries.html">Leibniz series</a> for <img src="/images/equations/PiFormulas/Inline25.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="50" height="21" alt="tan^(-1)x" />. The error after the <img src="/images/equations/PiFormulas/Inline26.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />th term of this series in the <a href="/GregorySeries.html">Gregory series</a> is larger than <img src="/images/equations/PiFormulas/Inline27.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="47" height="21" alt="(2n)^(-1)" /> so this sum converges so slowly that 300 terms are not sufficient to calculate <img src="/images/equations/PiFormulas/Inline28.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> correctly to two decimal places! However, it can be transformed to </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation2.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="161" height="51" alt=" pi=sum_(k=1)^infty(3^k-1)/(4^k)zeta(k+1), " /></td><td align="right" width="3"> <div id="eqn8" class="eqnum"> (8) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/PiFormulas/Inline29.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="30" height="21" alt="zeta(z)" /> is the <a href="/RiemannZetaFunction.html">Riemann zeta function</a> (Vardi 1991, pp. 157-158; Flajolet and Vardi 1996), so that the error after <img src="/images/equations/PiFormulas/Inline30.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" /> terms is <img src="/images/equations/PiFormulas/Inline31.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="65" height="21" alt=" approx (3/4)^k" />. </p> <p> An infinite sum series to Abraham Sharp (ca. 1717) is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation3.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="149" height="51" alt=" pi=sum_(k=0)^infty(2(-1)^k3^(1/2-k))/(2k+1) " /></td><td align="right" width="3"> <div id="eqn9" class="eqnum"> (9) </div> </td></tr> </table> </div> <p> (Smith 1953, p. 311). Additional simple series in which <img src="/images/equations/PiFormulas/Inline32.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> appears are </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline33.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="54" height="27" alt="1/4pisqrt(2)" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline34.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline35.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="164" height="51" alt="sum_(k=1)^(infty)[((-1)^(k+1))/(4k-1)+((-1)^(k+1))/(4k-3)]" /></td><td align="right" width="10"> <div id="eqn10" class="eqnum"> (10) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline36.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline37.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline38.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="187" height="26" alt="1+1/3-1/5-1/7+1/9+1/(11)-..." /></td><td align="right" width="10"> <div id="eqn11" class="eqnum"> (11) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline39.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="58" height="26" alt="1/4(pi-3)" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline40.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline41.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="166" height="51" alt="sum_(k=1)^(infty)((-1)^(k+1))/(2k(2k+1)(2k+2))" /></td><td align="right" width="10"> <div id="eqn12" class="eqnum"> (12) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline42.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline43.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline44.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="223" height="39" alt="1/(2·3·4)-1/(4·5·6)+1/(6·7·8)-..." /></td><td align="right" width="10"> <div id="eqn13" class="eqnum"> (13) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline45.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="26" alt="1/6pi^2" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline46.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline47.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="45" height="50" alt="sum_(k=1)^(infty)1/(k^2)" /></td><td align="right" width="10"> <div id="eqn14" class="eqnum"> (14) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline48.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline49.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline50.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="165" height="26" alt="1+1/4+1/9+1/(16)+1/(25)+..." /></td><td align="right" width="10"> <div id="eqn15" class="eqnum"> (15) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline51.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="30" height="26" alt="1/8pi^2" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline52.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline53.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="92" height="50" alt="sum_(k=1)^(infty)1/((2k-1)^2)" /></td><td align="right" width="10"> <div id="eqn16" class="eqnum"> (16) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline54.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline55.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline56.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="155" height="42" alt="1+1/(3^2)+1/(5^2)+1/(7^2)+..." /></td><td align="right" width="10"> <div id="eqn17" class="eqnum"> (17) </div> </td></tr> </table> </div> <p> (Wells 1986, p. 53). </p> <p> In 1666, Newton used a geometric construction to derive the formula </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline57.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="8" height="20" alt="pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline58.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline59.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="191" height="42" alt="3/4sqrt(3)+24int_0^(1/4)sqrt(x-x^2)dx" /></td><td align="right" width="10"> <div id="eqn18" class="eqnum"> (18) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline60.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline61.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline62.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="339" height="46" alt="(3sqrt(3))/4+24(1/(12)-1/(5·2^5)-1/(28·2^7)-1/(72·2^9)-...)," /></td><td align="right" width="10"> <div id="eqn19" class="eqnum"> (19) </div> </td></tr> </table> </div> <p> which he used to compute <img src="/images/equations/PiFormulas/Inline63.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> (Wells 1986, p. 50; Borwein <i>et al. </i>1989; Borwein and Bailey 2003, pp. 105-106). The coefficients can be found from the integral </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline64.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="29" height="20" alt="I(x)" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline65.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline66.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="97" height="38" alt="intsqrt(x-x^2)dx" /></td><td align="right" width="10"> <div id="eqn20" class="eqnum"> (20) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline67.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline68.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline69.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="254" height="29" alt="1/4(2x-1)sqrt(x-x^2)-1/8sin^(-1)(1-2x)" /></td><td align="right" width="10"> <div id="eqn21" class="eqnum"> (21) </div> </td></tr> </table> </div> <p> by taking the series expansion of <img src="/images/equations/PiFormulas/Inline70.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="81" height="21" alt="I(x)-I(0)" /> about 0, obtaining </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation4.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="371" height="26" alt=" I(x)=2/3x^(3/2)-1/5x^(5/2)-1/(28)x^(7/2)-1/(72)x^(9/2)-5/(704)x^(11/2)+... " /></td><td align="right" width="3"> <div id="eqn22" class="eqnum"> (22) </div> </td></tr> </table> </div> <p> (OEIS <a href="http://oeis.org/A054387">A054387</a> and <a href="http://oeis.org/A054388">A054388</a>). Using Euler's <a href="/ConvergenceImprovement.html">convergence improvement</a> transformation gives </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline71.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="34" alt="pi/2" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline72.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline73.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="233" height="50" alt="1/2sum_(n=0)^(infty)((n!)^22^(n+1))/((2n+1)!)=sum_(n=0)^(infty)(n!)/((2n+1)!!)" /></td><td align="right" width="10"> <div id="eqn23" class="eqnum"> (23) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline74.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline75.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline76.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="191" height="39" alt="1+1/3+(1·2)/(3·5)+(1·2·3)/(3·5·7)+..." /></td><td align="right" width="10"> <div id="eqn24" class="eqnum"> (24) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline77.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline78.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline79.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="259" height="40" alt="1+1/3(1+2/5(1+3/7(1+4/9(1+...))))" /></td><td align="right" width="10"> <div id="eqn25" class="eqnum"> (25) </div> </td></tr> </table> </div> <p> (Beeler <i>et al. </i>1972, Item 120). </p> <p> This corresponds to plugging <img src="/images/equations/PiFormulas/Inline80.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="82" height="27" alt="x=1/sqrt(2)" /> into the <a href="/PowerSeries.html">power series</a> for the <a href="/HypergeometricFunction.html">hypergeometric function</a> <img src="/images/equations/PiFormulas/Inline81.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="106" height="21" alt="_2F_1(a,b;c;x)" />, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation5.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="339" height="51" alt=" (sin^(-1)x)/(sqrt(1-x^2))=sum_(i=0)^infty((2x)^(2i+1)i!^2)/(2(2i+1)!)=_2F_1(1,1;3/2;x^2)x. " /></td><td align="right" width="3"> <div id="eqn26" class="eqnum"> (26) </div> </td></tr> </table> </div> <p> Despite the convergence improvement, series (◇) converges at only one bit/term. At the cost of a <a href="/SquareRoot.html">square root</a>, Gosper has noted that <img src="/images/equations/PiFormulas/Inline82.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="59" height="21" alt="x=1/2" /> gives 2 bits/term, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation6.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="184" height="50" alt=" 1/9sqrt(3)pi=1/2sum_(i=0)^infty((i!)^2)/((2i+1)!), " /></td><td align="right" width="3"> <div id="eqn27" class="eqnum"> (27) </div> </td></tr> </table> </div> <p> and <img src="/images/equations/PiFormulas/Inline83.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="103" height="21" alt="x=sin(pi/10)" /> gives almost 3.39 bits/term, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation7.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="237" height="51" alt=" pi/(5sqrt(phi+2))=1/2sum_(i=0)^infty((i!)^2)/(phi^(2i+1)(2i+1)!), " /></td><td align="right" width="3"> <div id="eqn28" class="eqnum"> (28) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/PiFormulas/Inline84.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="phi" /> is the <a href="/GoldenRatio.html">golden ratio</a>. Gosper also obtained </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation8.svg" data-src-small="/images/equations/PiFormulas/NumberedEquation8_400.svg" data-src-default="/images/equations/PiFormulas/NumberedEquation8.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="470" height="40" data-big="470 40" data-small="345 87" border="0" alt=" pi=3+1/(60)(8+(2·3)/(7·8·3)(13+(3·5)/(10·11·3)(18+(4·7)/(13·14·3)(23+...)))). " /></td><td align="right" width="3"> <div id="eqn29" class="eqnum"> (29) </div> </td></tr> </table> </div> <p> Various limits also converge to <img src="/images/equations/PiFormulas/Inline85.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" />, a simple example being </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation9.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="171" height="40" alt=" lim_(n->infty)ncos((pi(n-2))/(2n))=pi. " /></td><td align="right" width="3"> <div id="eqn30" class="eqnum"> (30) </div> </td></tr> </table> </div> <p> More interesting examples are given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation10.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="213" height="50" alt=" lim_(n->infty)((2(-1)^(n+1)(2n)!)/(2^(2n)B_(2n)))^(1/(2n))=pi, " /></td><td align="right" width="3"> <div id="eqn31" class="eqnum"> (31) </div> </td></tr> </table> </div> <p> and </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation11.svg" data-src-small="/images/equations/PiFormulas/NumberedEquation11_400.svg" data-src-default="/images/equations/PiFormulas/NumberedEquation11.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="407" height="50" data-big="407 50" data-small="374 78" border="0" alt=" lim_(n->infty)((2(-1)^(n+1)(2n)!)/(2^(2n)B_(2n)(1-2^(-n))(1-3^(-n))(1-5^(-n))(1-7^(-n))))^(1/(2n))=pi, " /></td><td align="right" width="3"> <div id="eqn32" class="eqnum"> (32) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/PiFormulas/Inline86.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="18" height="21" alt="B_n" /> is a <a href="/BernoulliNumber.html">Bernoulli number</a> (Plouffe 2022). These formulas can be used as a <a href="/Digit-ExtractionAlgorithm.html">digit-extraction algorithm</a> for <a href="/PiDigits.html">pi digits</a>. </p> <p> A <a href="/SpigotAlgorithm.html">spigot algorithm</a> for <img src="/images/equations/PiFormulas/Inline87.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> is given by Rabinowitz and Wagon (1995; Borwein and Bailey 2003, pp. 141-142). </p> <p> A closed form expression giving another <a href="/Digit-ExtractionAlgorithm.html">digit-extraction algorithm</a> which produces digits of <img src="/images/equations/PiFormulas/Inline88.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> (or <img src="/images/equations/PiFormulas/Inline89.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="pi^2" />) in base-16 was discovered by Bailey <i>et al. </i>(Bailey <i>et al. </i>1997, Adamchik and Wagon 1997), </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation12.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="370" height="49" alt=" pi=sum_(n=0)^infty(4/(8n+1)-2/(8n+4)-1/(8n+5)-1/(8n+6))(1/(16))^n. " /></td><td align="right" width="3"> <div id="eqn33" class="eqnum"> (33) </div> </td></tr> </table> </div> <p> This formula, known as the <a href="/BBPFormula.html">BBP formula</a>, was discovered using the <a href="/PSLQAlgorithm.html">PSLQ algorithm</a> (Ferguson <i>et al. </i>1999) and is equivalent to </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation13.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="208" height="46" alt=" pi=int_0^1(16y-16)/(y^4-2y^3+4y-4)dy. " /></td><td align="right" width="3"> <div id="eqn34" class="eqnum"> (34) </div> </td></tr> </table> </div> <p> There is a series of <a href="/BBP-TypeFormula.html">BBP-type formulas</a> for <img src="/images/equations/PiFormulas/Inline90.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> in powers of <img src="/images/equations/PiFormulas/Inline91.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="(-1)^k" />, the first few independent formulas of which are </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline92.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="8" height="20" alt="pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline93.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline94.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="87" height="51" alt="4sum_(k=0)^(infty)((-1)^k)/(2k+1)" /></td><td align="right" width="10"> <div id="eqn35" class="eqnum"> (35) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline95.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline96.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline97.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="206" height="50" alt="3sum_(k=0)^(infty)(-1)^k[1/(6k+1)+1/(6k+5)]" /></td><td align="right" width="10"> <div id="eqn36" class="eqnum"> (36) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline98.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline99.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline100.svg" data-src-small="/images/equations/PiFormulas/Inline100_400.svg" data-src-default="/images/equations/PiFormulas/Inline100.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="447" height="50" data-big="447 50" data-small="379 99" border="0" alt="4sum_(k=0)^(infty)(-1)^k[1/(10k+1)-1/(10k+3)+1/(10k+5)-1/(10k+7)+1/(10k+9)]" /></td><td align="right" width="10"> <div id="eqn37" class="eqnum"> (37) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline101.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline102.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline103.svg" data-src-small="/images/equations/PiFormulas/Inline103_400.svg" data-src-default="/images/equations/PiFormulas/Inline103.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="601" height="50" data-big="601 50" data-small="334 99" border="0" alt="sum_(k=0)^(infty)(-1)^k[3/(14k+1)-3/(14k+3)+3/(14k+5)+4/(14k+7)+4/(14k+9)-4/(14k+11)+4/(14k+13)]" /></td><td align="right" width="10"> <div id="eqn38" class="eqnum"> (38) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline104.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline105.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline106.svg" data-src-small="/images/equations/PiFormulas/Inline106_400.svg" data-src-default="/images/equations/PiFormulas/Inline106.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="692" height="50" data-big="692 50" data-small="367 99" border="0" alt="sum_(k=0)^(infty)(-1)^k[2/(18k+1)+3/(18k+3)+2/(18k+5)-2/(18k+7)-2/(18k+11)+2/(18k+13)+3/(18k+15)+2/(18k+17)]" /></td><td align="right" width="10"> <div id="eqn39" class="eqnum"> (39) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline107.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline108.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline109.svg" data-src-small="/images/equations/PiFormulas/Inline109_400.svg" data-src-default="/images/equations/PiFormulas/Inline109.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="937" height="50" data-big="937 50" data-small="354 146" border="0" alt="sum_(k=0)^(infty)(-1)^k[3/(22k+1)-3/(22k+3)+3/(22k+5)-3/(22k+7)+3/(22k+9)+8/(22k+11)+3/(22k+13)-3/(22k+15)+3/(22k+17)-3/(22k+19)+1/(22k+21)]." /></td><td align="right" width="10"> <div id="eqn40" class="eqnum"> (40) </div> </td></tr> </table> </div> <p> Similarly, there are a series of <a href="/BBP-TypeFormula.html">BBP-type formulas</a> for <img src="/images/equations/PiFormulas/Inline110.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> in powers of <img src="/images/equations/PiFormulas/Inline111.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="2^k" />, the first few independent formulas of which are </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline112.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="8" height="20" alt="pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline113.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline114.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="321" height="50" alt="sum_(k=0)^(infty)1/(16^k)[4/(8k+1)-2/(8k+4)-1/(8k+5)-1/(8k+6)]" /></td><td align="right" width="10"> <div id="eqn41" class="eqnum"> (41) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline115.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline116.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline117.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="339" height="50" alt="1/2sum_(k=0)^(infty)1/(16^k)[8/(8k+2)+4/(8k+3)+4/(8k+4)-1/(8k+7)]" /></td><td align="right" width="10"> <div id="eqn42" class="eqnum"> (42) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline118.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline119.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline120.svg" data-src-small="/images/equations/PiFormulas/Inline120_400.svg" data-src-default="/images/equations/PiFormulas/Inline120.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="712" height="50" data-big="712 50" data-small="342 146" border="0" alt="1/(16)sum_(k=0)^(infty)1/(256^k)[(64)/(16k+1)-(32)/(16k+4)-(16)/(16k+5)-(16)/(16k+6)+4/(16k+9)-2/(16k+12)-1/(16k+13)-1/(16k+14)]" /></td><td align="right" width="10"> <div id="eqn43" class="eqnum"> (43) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline121.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline122.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline123.svg" data-src-small="/images/equations/PiFormulas/Inline123_400.svg" data-src-default="/images/equations/PiFormulas/Inline123.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="720" height="50" data-big="720 50" data-small="350 146" border="0" alt="1/(32)sum_(k=0)^(infty)1/(256^k)[(128)/(16k+2)+(64)/(16k+3)+(64)/(16k+4)-(16)/(16k+7)+8/(16k+10)+4/(16k+11)+4/(16k+12)-1/(16k+15)]" /></td><td align="right" width="10"> <div id="eqn44" class="eqnum"> (44) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline124.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline125.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline126.svg" data-src-small="/images/equations/PiFormulas/Inline126_400.svg" data-src-default="/images/equations/PiFormulas/Inline126.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="969" height="50" data-big="969 50" data-small="350 146" border="0" alt="1/(32)sum_(k=0)^(infty)1/(4096^k)[(256)/(24k+2)+(192)/(24k+3)-(256)/(24k+4)-(96)/(24k+6)-(96)/(24k+8)+(16)/(24k+10)-4/(24k+12)-3/(24k+15)-6/(24k+16)-2/(24k+18)-1/(24k+20)]" /></td><td align="right" width="10"> <div id="eqn45" class="eqnum"> (45) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline127.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline128.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline129.svg" data-src-small="/images/equations/PiFormulas/Inline129_400.svg" data-src-default="/images/equations/PiFormulas/Inline129.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="1368" height="50" data-big="1368 50" data-small="357 240" border="0" alt="1/(64)sum_(k=0)^(infty)1/(4096^k)[(256)/(24k+1)+(256)/(24k+2)-(384)/(24k+3)-(256)/(24k+4)-(64)/(24k+5)+(96)/(24k+8)+(64)/(24k+9)+(16)/(24k+10)+8/(24k+12)-4/(24k+13)+6/(24k+15)+6/(24k+16)+1/(24k+17)+1/(24k+18)-1/(24k+20)-1/(24k+21)]" /></td><td align="right" width="10"> <div id="eqn46" class="eqnum"> (46) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline130.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline131.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline132.svg" data-src-small="/images/equations/PiFormulas/Inline132_400.svg" data-src-default="/images/equations/PiFormulas/Inline132.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="1368" height="50" data-big="1368 50" data-small="357 240" border="0" alt="1/(96)sum_(k=0)^(infty)1/(4096^k)[(256)/(24k+2)+(64)/(24k+3)+(128)/(24k+5)+(352)/(24k+6)+(64)/(24k+7)+(288)/(24k+8)+(128)/(24k+9)+(80)/(24k+10)+(20)/(24k+12)-(16)/(24k+14)-1/(24k+15)+6/(24k+16)-2/(24k+17)-1/(24k+19)+1/(24k+20)-2/(24k+21)]" /></td><td align="right" width="10"> <div id="eqn47" class="eqnum"> (47) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline133.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline134.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline135.svg" data-src-small="/images/equations/PiFormulas/Inline135_400.svg" data-src-default="/images/equations/PiFormulas/Inline135.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="1443" height="50" data-big="1443 50" data-small="386 240" border="0" alt="1/(96)sum_(k=0)^(infty)1/(4096^k)[(256)/(24k+1)+(320)/(24k+3)+(256)/(24k+4)-(192)/(24k+5)-(224)/(24k+6)-(64)/(24k+7)-(192)/(24k+8)-(64)/(24k+9)-(64)/(24k+10)-(28)/(24k+12)-4/(24k+13)-5/(24k+15)+3/(24k+17)+1/(24k+18)+1/(24k+19)+1/(24k+21)-1/(24k+22)]" /></td><td align="right" width="10"> <div id="eqn48" class="eqnum"> (48) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline136.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline137.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline138.svg" data-src-small="/images/equations/PiFormulas/Inline138_400.svg" data-src-default="/images/equations/PiFormulas/Inline138.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="1526" height="50" data-big="1526 50" data-small="357 240" border="0" alt="1/(96)sum_(k=0)^(infty)1/(4096^k)[(512)/(24k+1)-(256)/(24k+2)+(64)/(24k+3)-(512)/(24k+4)-(32)/(24k+6)+(64)/(24k+7)+(96)/(24k+8)+(64)/(24k+9)+(48)/(24k+10)-(12)/(24k+12)-8/(24k+13)-(16)/(24k+14)-1/(24k+15)-6/(24k+16)-2/(24k+18)-1/(24k+19)-1/(24k+20)-1/(24k+21)]" /></td><td align="right" width="10"> <div id="eqn49" class="eqnum"> (49) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline139.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline140.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline141.svg" data-src-small="/images/equations/PiFormulas/Inline141_400.svg" data-src-default="/images/equations/PiFormulas/Inline141.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="1412" height="50" data-big="1412 50" data-small="357 240" border="0" alt="1/(4096)sum_(k=0)^(infty)1/(65536^k)[(16384)/(32k+1)-(8192)/(32k+4)-(4096)/(32k+5)-(4096)/(32k+6)+(1024)/(32k+9)-(512)/(32k+12)-(256)/(32k+13)-(256)/(32k+14)+(64)/(32k+17)-(32)/(32k+20)-(16)/(32k+21)-(16)/(32k+22)+4/(32k+25)-2/(32k+28)-1/(32k+29)-1/(32k+30)]" /></td><td align="right" width="10"> <div id="eqn50" class="eqnum"> (50) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline142.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline143.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline144.svg" data-src-small="/images/equations/PiFormulas/Inline144_400.svg" data-src-default="/images/equations/PiFormulas/Inline144.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="1424" height="50" data-big="1424 50" data-small="357 240" border="0" alt="1/(4096)sum_(k=0)^(infty)1/(65536^k)[(32768)/(32k+2)+(16384)/(32k+3)+(16384)/(32k+4)-(4096)/(32k+7)+(2048)/(32k+10)+(1024)/(32k+11)+(1024)/(32k+12)-(256)/(32k+15)+(128)/(32k+18)+(64)/(32k+19)+(64)/(32k+20)-(16)/(32k+23)+8/(32k+26)+4/(32k+27)+4/(32k+28)-1/(32k+31)]." /></td><td align="right" width="10"> <div id="eqn51" class="eqnum"> (51) </div> </td></tr> </table> </div> <p> F. Bellard found the rapidly converging <a href="/BBP-TypeFormula.html">BBP-type formula</a> </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation14.svg" data-src-small="/images/equations/PiFormulas/NumberedEquation14_400.svg" data-src-default="/images/equations/PiFormulas/NumberedEquation14.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="651" height="50" data-big="651 50" data-small="316 164" border="0" alt=" pi=1/(2^6)sum_(n=0)^infty((-1)^n)/(2^(10n))(-(2^5)/(4n+1)-1/(4n+3)+(2^8)/(10n+1)-(2^6)/(10n+3)-(2^2)/(10n+5)-(2^2)/(10n+7)+1/(10n+9)). " /></td><td align="right" width="3"> <div id="eqn52" class="eqnum"> (52) </div> </td></tr> </table> </div> <p> A related integral is </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation15.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="188" height="45" alt=" pi=(22)/7-int_0^1(x^4(1-x)^4)/(1+x^2)dx " /></td><td align="right" width="3"> <div id="eqn53" class="eqnum"> (53) </div> </td></tr> </table> </div> <p> (Dalzell 1944, 1971; Le Lionnais 1983, p. 22; Borwein, Bailey, and Girgensohn 2004, p. 3; Boros and Moll 2004, p. 125; Lucas 2005; Borwein <i>et al. </i>2007, p. 14). This integral was known by K. Mahler in the mid-1960s and appears in an exam at the University of Sydney in November 1960 (Borwein, Bailey, and Girgensohn, p. 3). Beukers (2000) and Boros and Moll (2004, p. 126) state that it is not clear if these exists a natural choice of rational polynomial whose integral between 0 and 1 produces <img src="/images/equations/PiFormulas/Inline145.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="93" height="21" alt="pi-333/106" />, where 333/106 is the next convergent. However, an integral exists for the <i>fourth</i> convergent, namely </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation16.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="334" height="47" alt=" pi=(355)/(113)-1/(3164)int_0^1(x^8(1-x)^8(25+816x^2))/(1+x^2)dx. " /></td><td align="right" width="3"> <div id="eqn54" class="eqnum"> (54) </div> </td></tr> </table> </div> <p> (Lucas 2005; Bailey <i>et al. </i>2007, p. 219). In fact, Lucas (2005) gives a few other such integrals. </p> <p> Backhouse (1995) used the identity </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline146.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="26" height="22" alt="I_(m,n)" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline147.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline148.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="123" height="43" alt="int_0^1(x^m(1-x)^n)/(1+x^2)dx" /></td><td align="right" width="10"> <div id="eqn55" class="eqnum"> (55) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline149.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline150.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline151.svg" data-src-small="/images/equations/PiFormulas/Inline151_400.svg" data-src-default="/images/equations/PiFormulas/Inline151.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="566" height="40" data-big="566 40" data-small="322 68" border="0" alt="2^(-(m+n+1))sqrt(pi)Gamma(m+1)Gamma(n+1)×_3F_2(1,(m+1)/2,(m+2)/2;(m+n+2)/2,(m+n+3)/2;-1)" /></td><td align="right" width="10"> <div id="eqn56" class="eqnum"> (56) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline152.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline153.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline154.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="97" height="20" alt="a+bpi+cln2" /></td><td align="right" width="10"> <div id="eqn57" class="eqnum"> (57) </div> </td></tr> </table> </div> <p> for positive integer <img src="/images/equations/PiFormulas/Inline155.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="m" /> and <img src="/images/equations/PiFormulas/Inline156.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> and where <img src="/images/equations/PiFormulas/Inline157.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="a" />, <img src="/images/equations/PiFormulas/Inline158.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="b" />, and <img src="/images/equations/PiFormulas/Inline159.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="c" /> are rational constant to generate a number of formulas for <img src="/images/equations/PiFormulas/Inline160.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" />. In particular, if <img src="/images/equations/PiFormulas/Inline161.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="141" height="21" alt="2m-n=0 (mod 4)" />, then <img src="/images/equations/PiFormulas/Inline162.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="c=0" /> (Lucas 2005). </p> <p> A similar formula was subsequently discovered by Ferguson, leading to a two-dimensional lattice of such formulas which can be generated by these two formulas given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation17.svg" data-src-small="/images/equations/PiFormulas/NumberedEquation17_400.svg" data-src-default="/images/equations/PiFormulas/NumberedEquation17.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="563" height="50" data-big="563 50" data-small="345 101" border="0" alt=" pi=sum_(k=0)^infty((4+8r)/(8k+1)-(8r)/(8k+2)-(4r)/(8k+3)-(2+8r)/(8k+4)-(1+2r)/(8k+5)-(1+2r)/(8k+6)+r/(8k+7))(1/(16))^k " /></td><td align="right" width="3"> <div id="eqn58" class="eqnum"> (58) </div> </td></tr> </table> </div> <p> for any complex value of <img src="/images/equations/PiFormulas/Inline163.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="7" height="21" alt="r" /> (Adamchik and Wagon), giving the <a href="/BBPFormula.html">BBP formula</a> as the special case <img src="/images/equations/PiFormulas/Inline164.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="37" height="21" alt="r=0" />. </p> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="255.607" height="255.596" src="images/eps-svg/PiFormulasWagonIdentity_700.svg" class="" alt="PiFormulasWagonIdentity" /> </div> <p> An even more general identity due to Wagon is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation18.svg" data-src-small="/images/equations/PiFormulas/NumberedEquation18_400.svg" data-src-default="/images/equations/PiFormulas/NumberedEquation18.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="686" height="51" data-big="686 51" data-small="282 165" border="0" alt=" pi+4tan^(-1)z+2ln((1-2z-z^2)/(z^2+1))=sum_(k=0)^infty1/(16^k)[(4(z+1)^(8k+1))/(8k+1)-(2(z+1)^(8k+4))/(8k+4)-((z+1)^(8k+5))/(8k+5)-((z+1)^(8k+6))/(8k+6)] " /></td><td align="right" width="3"> <div id="eqn59" class="eqnum"> (59) </div> </td></tr> </table> </div> <p> (Borwein and Bailey 2003, p. 141), which holds over a region of the <a href="/ComplexPlane.html">complex plane</a> excluding two triangular portions symmetrically placed about the <a href="/RealAxis.html">real axis</a>, as illustrated above. </p> <p> A perhaps even stranger general class of identities is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation19.svg" data-src-small="/images/equations/PiFormulas/NumberedEquation19_400.svg" data-src-default="/images/equations/PiFormulas/NumberedEquation19.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="859" height="53" data-big="859 53" data-small="378 167" border="0" alt=" pi=4sum_(j=1)^n((-1)^(j+1))/(2j-1)+((-1)^n(2n-1)!)/4sum_(k=0)^infty1/(16^k)[8/((8k+1)_(2n))-4/((8k+3)_(2n))-4/((8k+4)_(2n))-2/((8k+5)_(2n))+1/((8k+7)_(2n))+1/((8k+8)_(2n))] " /></td><td align="right" width="3"> <div id="eqn60" class="eqnum"> (60) </div> </td></tr> </table> </div> <p> which holds for any positive integer <img src="/images/equations/PiFormulas/Inline165.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />, where <img src="/images/equations/PiFormulas/Inline166.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="26" height="22" alt="(x)_n" /> is a <a href="/PochhammerSymbol.html">Pochhammer symbol</a> (B. Cloitre, pers. comm., Jan. 23, 2005). Even more amazingly, there is a closely analogous formula for the <a href="/NaturalLogarithmof2.html">natural logarithm of 2</a>. </p> <p> Following the discovery of the base-16 digit <a href="/BBPFormula.html">BBP formula</a> and related formulas, similar formulas in other bases were investigated. Borwein, Bailey, and Girgensohn (2004) have recently shown that <img src="/images/equations/PiFormulas/Inline167.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> has no Machin-type BBP arctangent formula that is not binary, although this does not rule out a completely different scheme for <a href="/Digit-ExtractionAlgorithm.html">digit-extraction algorithms</a> in other bases. </p> <p> S. Plouffe has devised an algorithm to compute the <img src="/images/equations/PiFormulas/Inline168.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />th <a href="/Digit.html">digit</a> of <img src="/images/equations/PiFormulas/Inline169.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> in any base in <img src="/images/equations/PiFormulas/Inline170.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="100" height="25" alt="O(n^3(logn)^3)" /> steps. </p> <p> A slew of additional identities due to Ramanujan, Catalan, and Newton are given by Castellanos (1988ab, pp. 86-88), including several involving sums of <a href="/FibonacciNumber.html">Fibonacci numbers</a>. Ramanujan found </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation20.svg" data-src-small="/images/equations/PiFormulas/NumberedEquation20_400.svg" data-src-default="/images/equations/PiFormulas/NumberedEquation20.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="473" height="58" data-big="473 58" data-small="253 121" border="0" alt=" sum_(k=0)^infty((-1)^k(4k+1)[(2k-1)!!]^3)/([(2k)!!]^3)=sum_(k=0)^infty((-1)^k(4k+1)[Gamma(k+1/2)]^3)/(pi^(3/2)[Gamma(k+1)]^3)=2/pi " /></td><td align="right" width="3"> <div id="eqn61" class="eqnum"> (61) </div> </td></tr> </table> </div> <p> (Hardy 1923, 1924, 1999, p. 7). </p> <p> Plouffe (2006) found the beautiful formula </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation21.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="296" height="108" alt=" pi=72sum_(n=1)^infty1/(n(e^(npi)-1))-96sum_(n=1)^infty1/(n(e^(2npi)-1)) +24sum_(n=1)^infty1/(n(e^(4npi)-1)). " /></td><td align="right" width="3"> <div id="eqn62" class="eqnum"> (62) </div> </td></tr> </table> </div> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="380.4" height="237.75" src="images/eps-svg/PiBlatnerProduct_1000.svg" class="" alt="PiBlatnerProduct" /> </div> <p> An interesting <a href="/InfiniteProduct.html">infinite product</a> formula due to Euler which relates <img src="/images/equations/PiFormulas/Inline171.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> and the <img src="/images/equations/PiFormulas/Inline172.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />th <a href="/PrimeNumber.html">prime</a> <img src="/images/equations/PiFormulas/Inline173.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="18" height="22" alt="p_n" /> is </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline174.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="8" height="20" alt="pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline175.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline176.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="142" height="68" alt="2/(product_(n=1)^(infty)[1+(sin(1/2pip_n))/(p_n)])" /></td><td align="right" width="10"> <div id="eqn63" class="eqnum"> (63) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline177.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline178.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline179.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="144" height="57" alt="2/(product_(n=2)^(infty)[1+((-1)^((p_n-1)/2))/(p_n)])" /></td><td align="right" width="10"> <div id="eqn64" class="eqnum"> (64) </div> </td></tr> </table> </div> <p> (Blatner 1997, p. 119), plotted above as a function of the number of terms in the product. </p> <p> A method similar to Archimedes' can be used to estimate <img src="/images/equations/PiFormulas/Inline180.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> by starting with an <img src="/images/equations/PiFormulas/Inline181.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />-gon and then relating the <a href="/Area.html">area</a> of subsequent <img src="/images/equations/PiFormulas/Inline182.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="2n" />-gons. Let <img src="/images/equations/PiFormulas/Inline183.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="10" height="21" alt="beta" /> be the <a href="/Angle.html">angle</a> from the center of one of the <a href="/Polygon.html">polygon</a>'s segments, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation22.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="105" height="26" alt=" beta=1/4(n-3)pi, " /></td><td align="right" width="3"> <div id="eqn65" class="eqnum"> (65) </div> </td></tr> </table> </div> <p> then </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation23.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="191" height="47" alt=" pi=(2sin(2beta))/((n-3)product_(k=0)^(infty)cos(2^(-k)beta)) " /></td><td align="right" width="3"> <div id="eqn66" class="eqnum"> (66) </div> </td></tr> </table> </div> <p> (Beckmann 1989, pp. 92-94). </p> <p> Vieta (1593) was the first to give an exact expression for <img src="/images/equations/PiFormulas/Inline184.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> by taking <img src="/images/equations/PiFormulas/Inline185.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=4" /> in the above expression, giving </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation24.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="212" height="46" alt=" cosbeta=sinbeta=1/(sqrt(2))=1/2sqrt(2), " /></td><td align="right" width="3"> <div id="eqn67" class="eqnum"> (67) </div> </td></tr> </table> </div> <p> which leads to an <a href="/InfiniteProduct.html">infinite product</a> of <a href="/NestedRadical.html">nested radicals</a>, </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation25.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="345" height="47" alt=" 2/pi=sqrt(1/2)sqrt(1/2+1/2sqrt(1/2))sqrt(1/2+1/2sqrt(1/2+1/2sqrt(1/2)))... " /></td><td align="right" width="3"> <div id="eqn68" class="eqnum"> (68) </div> </td></tr> </table> </div> <p> (Wells 1986, p. 50; Beckmann 1989, p. 95). However, this expression was not rigorously proved to converge until Rudio in 1892. </p> <p> A related formula is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation26.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="332" height="70" alt=" pi=lim_(n->infty)2^(n+1)sqrt(2-sqrt(2+sqrt(2+sqrt(2+...+sqrt(2))))_()_(n)), " /></td><td align="right" width="3"> <div id="eqn69" class="eqnum"> (69) </div> </td></tr> </table> </div> <p> which can be written </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation27.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="107" height="30" alt=" pi=lim_(n->infty)2^(n+1)pi_n, " /></td><td align="right" width="3"> <div id="eqn70" class="eqnum"> (70) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/PiFormulas/Inline186.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="pi_n" /> is defined using the iteration </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation28.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="282" height="47" alt=" pi_n=sqrt((1/2pi_(n-1))^2+[1-sqrt(1-(1/2pi_(n-1))^2)]^2) " /></td><td align="right" width="3"> <div id="eqn71" class="eqnum"> (71) </div> </td></tr> </table> </div> <p> with <img src="/images/equations/PiFormulas/Inline187.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="66" height="24" alt="pi_0=sqrt(2)" /> (J. Munkhammar, pers. comm., April 27, 2000). The formula </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation29.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="381" height="62" alt=" pi=2lim_(m->infty)sum_(n=1)^msqrt([sqrt(1-((n-1)/m)^2)-sqrt(1-(n/m)^2)]^2+1/(m^2)) " /></td><td align="right" width="3"> <div id="eqn72" class="eqnum"> (72) </div> </td></tr> </table> </div> <p> is also closely related. </p> <p> A pretty formula for <img src="/images/equations/PiFormulas/Inline188.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation30.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="150" height="64" alt=" pi=(product_(n=1)^(infty)(1+1/(4n^2-1)))/(sum_(n=1)^(infty)1/(4n^2-1)), " /></td><td align="right" width="3"> <div id="eqn73" class="eqnum"> (73) </div> </td></tr> </table> </div> <p> where the numerator is a form of the <a href="/WallisFormula.html">Wallis formula</a> for <img src="/images/equations/PiFormulas/Inline189.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="pi/2" /> and the denominator is a <a href="/TelescopingSum.html">telescoping sum</a> with sum 1/2 since </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation31.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="230" height="42" alt=" 1/(4n^2-1)=1/2(1/(2n-1)-1/(2n+1)) " /></td><td align="right" width="3"> <div id="eqn74" class="eqnum"> (74) </div> </td></tr> </table> </div> <p> (Sondow 1997). </p> <p> A particular case of the <a href="/WallisFormula.html">Wallis formula</a> gives </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation32.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="341" height="50" alt=" pi/2=product_(n=1)^infty[((2n)^2)/((2n-1)(2n+1))]=(2·2)/(1·3)(4·4)/(3·5)(6·6)/(5·7)... " /></td><td align="right" width="3"> <div id="eqn75" class="eqnum"> (75) </div> </td></tr> </table> </div> <p> (Wells 1986, p. 50). This formula can also be written </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation33.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="265" height="71" alt=" lim_(n->infty)(2^(4n))/(n(2n; n)^2)=pilim_(n->infty)(n[Gamma(n)]^2)/([Gamma(1/2+n)]^2)=pi, " /></td><td align="right" width="3"> <div id="eqn76" class="eqnum"> (76) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/PiFormulas/Inline190.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="43" alt="(n; k)" /> denotes a <a href="/BinomialCoefficient.html">binomial coefficient</a> and <img src="/images/equations/PiFormulas/Inline191.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="33" height="21" alt="Gamma(x)" /> is the <a href="/GammaFunction.html">gamma function</a> (Knopp 1990). Euler obtained </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation34.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="234" height="50" alt=" pi=sqrt(6(1+1/(2^2)+1/(3^2)+1/(4^2)+...)), " /></td><td align="right" width="3"> <div id="eqn77" class="eqnum"> (77) </div> </td></tr> </table> </div> <p> which follows from the special value of the <a href="/RiemannZetaFunction.html">Riemann zeta function</a> <img src="/images/equations/PiFormulas/Inline192.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="91" height="25" alt="zeta(2)=pi^2/6" />. Similar <a href="/Formula.html">formulas</a> follow from <img src="/images/equations/PiFormulas/Inline193.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="44" height="21" alt="zeta(2n)" /> for all <a href="/PositiveInteger.html">positive integers</a> <img src="/images/equations/PiFormulas/Inline194.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" />. </p> <p> An infinite sum due to Ramanujan is </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation35.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="169" height="49" alt=" 1/pi=sum_(n=0)^infty(2n; n)^3(42n+5)/(2^(12n+4)) " /></td><td align="right" width="3"> <div id="eqn78" class="eqnum"> (78) </div> </td></tr> </table> </div> <p> (Borwein <i>et al. </i>1989; Borwein and Bailey 2003, p. 109; Bailey <i>et al. </i>2007, p. 44). Further sums are given in Ramanujan (1913-14), </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation36.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="367" height="49" alt=" 4/pi=sum_(n=0)^infty((-1)^n(1123+21460n)(2n-1)!!(4n-1)!!)/(882^(2n+1)32^n(n!)^3) " /></td><td align="right" width="3"> <div id="eqn79" class="eqnum"> (79) </div> </td></tr> </table> </div> <p> and </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline195.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="39" alt="1/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline196.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline197.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="322" height="49" alt="sqrt(8)sum_(n=0)^(infty)((1103+26390n)(2n-1)!!(4n-1)!!)/(99^(4n+2)32^n(n!)^3)" /></td><td align="right" width="10"> <div id="eqn80" class="eqnum"> (80) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline198.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline199.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline200.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="231" height="52" alt="(sqrt(8))/(9801)sum_(n=0)^(infty)((4n)!(1103+26390n))/((n!)^4396^(4n))" /></td><td align="right" width="10"> <div id="eqn81" class="eqnum"> (81) </div> </td></tr> </table> </div> <p> (Beeler <i>et al. </i>1972, Item 139; Borwein <i>et al. </i>1989; Borwein and Bailey 2003, p. 108; Bailey <i>et al. </i>2007, p. 44). Equation (<a href="#eqn81">81</a>) is derived from a modular identity of order 58, although a first derivation was not presented prior to Borwein and Borwein (1987). The above series both give </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation37.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="242" height="46" alt=" pi approx (9801)/(2206sqrt(2))=3.14159273001... " /></td><td align="right" width="3"> <div id="eqn82" class="eqnum"> (82) </div> </td></tr> </table> </div> <p> (Wells 1986, p. 54) as the first approximation and provide, respectively, about 6 and 8 decimal places per term. Such series exist because of the rationality of various modular invariants. </p> <p> The general form of the series is </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation38.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="337" height="53" alt=" sum_(n=0)^infty[a(t)+nb(t)]((6n)!)/((3n)!(n!)^3)1/([j(t)]^n)=(sqrt(-j(t)))/pi, " /></td><td align="right" width="3"> <div id="eqn83" class="eqnum"> (83) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/PiFormulas/Inline201.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="5" height="21" alt="t" /> is a <a href="/BinaryQuadraticFormDiscriminant.html">binary quadratic form discriminant</a>, <img src="/images/equations/PiFormulas/Inline202.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="27" height="21" alt="j(t)" /> is the <a href="/j-Function.html"><i>j</i>-function</a>, </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline203.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="27" height="20" alt="b(t)" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline204.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline205.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="113" height="25" alt="sqrt(t[1728-j(t)])" /></td><td align="right" width="10"> <div id="eqn84" class="eqnum"> (84) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline206.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="27" height="20" alt="a(t)" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline207.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline208.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="232" height="49" alt="(b(t))/6{1-(E_4(t))/(E_6(t))[E_2(t)-6/(pisqrt(t))]}," /></td><td align="right" width="10"> <div id="eqn85" class="eqnum"> (85) </div> </td></tr> </table> </div> <p> and the <img src="/images/equations/PiFormulas/Inline209.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="17" height="21" alt="E_i" /> are <a href="/EisensteinSeries.html">Eisenstein series</a>. A <a href="/ClassNumber.html">class number</a> <img src="/images/equations/PiFormulas/Inline210.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="p" /> field involves <img src="/images/equations/PiFormulas/Inline211.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="p" />th degree <a href="/AlgebraicInteger.html">algebraic integers</a> of the constants <img src="/images/equations/PiFormulas/Inline212.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="61" height="21" alt="A=a(t)" />, <img src="/images/equations/PiFormulas/Inline213.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="60" height="21" alt="B=b(t)" />, and <img src="/images/equations/PiFormulas/Inline214.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="61" height="21" alt="C=c(t)" />. Of all series consisting of only integer terms, the one gives the most numeric digits in the shortest period of time corresponds to the largest <a href="/ClassNumber.html">class number</a> 1 discriminant of <img src="/images/equations/PiFormulas/Inline215.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="70" height="21" alt="d=-163" /> and was formulated by the Chudnovsky brothers (1987). The 163 appearing here is the same one appearing in the fact that <img src="/images/equations/PiFormulas/Inline216.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="51" height="23" alt="e^(pisqrt(163))" /> (the <a href="/RamanujanConstant.html">Ramanujan constant</a>) is very nearly an <a href="/AlmostInteger.html">integer</a>. Similarly, the factor <img src="/images/equations/PiFormulas/Inline217.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="64" height="21" alt="640320^3" /> comes from the <a href="/j-Function.html"><i>j</i>-function</a> identity for <img src="/images/equations/PiFormulas/Inline218.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="132" height="28" alt="j(1/2(1+isqrt(163)))" />. The series is given by </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline219.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="39" alt="1/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline220.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline221.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="324" height="53" alt="12sum_(n=0)^(infty)((-1)^n(6n)!(13591409+545140134n))/((n!)^3(3n)!(640320^3)^(n+1/2))" /></td><td align="right" width="10"> <div id="eqn86" class="eqnum"> (86) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline222.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline223.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline224.svg" data-src-small="/images/equations/PiFormulas/Inline224_400.svg" data-src-default="/images/equations/PiFormulas/Inline224.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="591" height="49" data-big="591 49" data-small="257 154" border="0" alt="(163·8·27·7·11·19·127)/(640320^(3/2))sum_(n=0)^(infty)((13591409)/(163·2·9·7·11·19·127)+n)((6n)!)/((3n)!(n!)^3)((-1)^n)/(640320^(3n))" /></td><td align="right" width="10"> <div id="eqn87" class="eqnum"> (87) </div> </td></tr> </table> </div> <p> (Borwein and Borwein 1993; Beck and Trott; Bailey <i>et al. </i>2007, p. 44). This series gives 14 digits accurately per term. The same equation in another form was given by the Chudnovsky brothers (1987) and is used by the <a href="http://www.wolfram.com/language/">Wolfram Language</a> to calculate <img src="/images/equations/PiFormulas/Inline225.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> (Vardi 1991; Wolfram Research), </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation39.svg" data-src-small="/images/equations/PiFormulas/NumberedEquation39_400.svg" data-src-default="/images/equations/PiFormulas/NumberedEquation39.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="402" height="55" data-big="402 55" data-small="387 83" border="0" alt=" pi=(426880sqrt(10005))/(A[_3F_2(1/6,1/2,5/6;1,1;B)-C_3F_2(7/6,3/2,(11)/6;2,2;B)]), " /></td><td align="right" width="3"> <div id="eqn88" class="eqnum"> (88) </div> </td></tr> </table> </div> <p> where </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline226.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="11" height="20" alt="A" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline227.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline228.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="73" height="20" alt="13591409" /></td><td align="right" width="10"> <div id="eqn89" class="eqnum"> (89) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline229.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="11" height="20" alt="B" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline230.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline231.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="115" height="26" alt="-1/(151931373056000)" /></td><td align="right" width="10"> <div id="eqn90" class="eqnum"> (90) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline232.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="C" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline233.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline234.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="155" height="26" alt="(30285563)/(1651969144908540723200)." /></td><td align="right" width="10"> <div id="eqn91" class="eqnum"> (91) </div> </td></tr> </table> </div> <p> The best formula for <a href="/ClassNumber.html">class number</a> 2 (largest discriminant <img src="/images/equations/PiFormulas/Inline235.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="-427" />) is </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation40.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="231" height="49" alt=" 1/pi=12sum_(n=0)^infty((-1)^n(6n)!(A+Bn))/((n!)^3(3n)!C^(n+1/2)), " /></td><td align="right" width="3"> <div id="eqn92" class="eqnum"> (92) </div> </td></tr> </table> </div> <p> where </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline236.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="11" height="20" alt="A" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline237.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline238.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="286" height="22" alt="212175710912sqrt(61)+1657145277365" /></td><td align="right" width="10"> <div id="eqn93" class="eqnum"> (93) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline239.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="11" height="20" alt="B" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline240.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline241.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="322" height="22" alt="13773980892672sqrt(61)+107578229802750" /></td><td align="right" width="10"> <div id="eqn94" class="eqnum"> (94) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline242.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="C" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline243.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline244.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="217" height="28" alt="[5280(236674+30303sqrt(61))]^3" /></td><td align="right" width="10"> <div id="eqn95" class="eqnum"> (95) </div> </td></tr> </table> </div> <p> (Borwein and Borwein 1993). This series adds about 25 digits for each additional term. The fastest converging series for <a href="/ClassNumber.html">class number</a> 3 corresponds to <img src="/images/equations/PiFormulas/Inline245.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="70" height="21" alt="d=-907" /> and gives 37-38 digits per term. The fastest converging <a href="/ClassNumber.html">class number</a> 4 series corresponds to <img src="/images/equations/PiFormulas/Inline246.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="79" height="21" alt="d=-1555" /> and is </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation41.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="242" height="54" alt=" (sqrt(-C^3))/pi=sum_(n=0)^infty((6n)!)/((3n)!(n!)^3)(A+nB)/(C^(3n)), " /></td><td align="right" width="3"> <div id="eqn96" class="eqnum"> (96) </div> </td></tr> </table> </div> <p> where </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline247.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="11" height="20" alt="A" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline248.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline249.svg" data-src-small="/images/equations/PiFormulas/Inline249_400.svg" data-src-default="/images/equations/PiFormulas/Inline249.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="1542" height="28" data-big="1542 28" data-small="384 155" border="0" alt="63365028312971999585426220+28337702140800842046825600sqrt(5)+384sqrt(5)(10891728551171178200467436212395209160385656017+4870929086578810225077338534541688721351255040sqrt(5))^(1/2)" /></td><td align="right" width="10"> <div id="eqn97" class="eqnum"> (97) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline250.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="11" height="20" alt="B" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline251.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline252.svg" data-src-small="/images/equations/PiFormulas/Inline252_400.svg" data-src-default="/images/equations/PiFormulas/Inline252.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="1526" height="28" data-big="1526 28" data-small="375 179" border="0" alt="7849910453496627210289749000+3510586678260932028965606400sqrt(5)+2515968sqrt(3110)(6260208323789001636993322654444020882161+2799650273060444296577206890718825190235sqrt(5))^(1/2)" /></td><td align="right" width="10"> <div id="eqn98" class="eqnum"> (98) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline253.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="12" height="20" alt="C" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline254.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline255.svg" data-src-small="/images/equations/PiFormulas/Inline255_400.svg" data-src-default="/images/equations/PiFormulas/Inline255.svg" class="displayformula swappable" style="max-height:100%;max-width:100%" width="1070" height="28" data-big="1070 28" data-small="371 106" border="0" alt="-214772995063512240-96049403338648032sqrt(5)-1296sqrt(5)(10985234579463550323713318473+4912746253692362754607395912sqrt(5))^(1/2)." /></td><td align="right" width="10"> <div id="eqn99" class="eqnum"> (99) </div> </td></tr> </table> </div> <p> This gives 50 digits per term. Borwein and Borwein (1993) have developed a general <a href="/Algorithm.html">algorithm</a> for generating such series for arbitrary <a href="/ClassNumber.html">class number</a>. </p> <p> A complete listing of Ramanujan's series for <img src="/images/equations/PiFormulas/Inline256.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="1/pi" /> found in his second and third notebooks is given by Berndt (1994, pp. 352-354), </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline257.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="39" alt="4/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline258.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline259.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="118" height="57" alt="sum_(n=0)^(infty)((6n+1)(1/2)_n^3)/(4^n(n!)^3)" /></td><td align="right" width="10"> <div id="eqn100" class="eqnum"> (100) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline260.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="23" height="39" alt="(16)/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline261.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline262.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="126" height="57" alt="sum_(n=0)^(infty)((42n+5)(1/2)_n^3)/(64^n(n!)^3)" /></td><td align="right" width="10"> <div id="eqn101" class="eqnum"> (101) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline263.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="23" height="39" alt="(32)/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline264.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline265.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="344" height="57" alt="sum_(n=0)^(infty)((42sqrt(5)n+5sqrt(5)+30n-1)(1/2)_n^3)/(64^n(n!)^3)((sqrt(5)-1)/2)^(8n)" /></td><td align="right" width="10"> <div id="eqn102" class="eqnum"> (102) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline266.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="26" height="39" alt="(27)/(4pi)" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline267.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline268.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="235" height="55" alt="sum_(n=0)^(infty)((15n+2)(1/2)_n(1/3)_n(2/3)_n)/((n!)^3)(2/(27))^n" /></td><td align="right" width="10"> <div id="eqn103" class="eqnum"> (103) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline269.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="53" height="44" alt="(15sqrt(3))/(2pi)" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline270.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline271.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="244" height="55" alt="sum_(n=0)^(infty)((33n+4)(1/2)_n(1/3)_n(2/3)_n)/((n!)^3)(4/(125))^n" /></td><td align="right" width="10"> <div id="eqn104" class="eqnum"> (104) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline272.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="56" height="51" alt="(5sqrt(5))/(2pisqrt(3))" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline273.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline274.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="244" height="55" alt="sum_(n=0)^(infty)((11n+1)(1/2)_n(1/6)_n(5/6)_n)/((n!)^3)(4/(125))^n" /></td><td align="right" width="10"> <div id="eqn105" class="eqnum"> (105) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline275.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="64" height="51" alt="(85sqrt(85))/(18pisqrt(3))" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline276.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline277.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="252" height="55" alt="sum_(n=0)^(infty)((133n+8)(1/2)_n(1/6)_n(5/6)_n)/((n!)^3)(4/(85))^(3n)" /></td><td align="right" width="10"> <div id="eqn106" class="eqnum"> (106) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline278.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="39" alt="4/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline279.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline280.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="230" height="55" alt="sum_(n=0)^(infty)((-1)^n(20n+3)(1/2)_n(1/4)_n(3/4)_n)/((n!)^32^(2n+1))" /></td><td align="right" width="10"> <div id="eqn107" class="eqnum"> (107) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline281.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="44" height="46" alt="4/(pisqrt(3))" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline282.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline283.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="230" height="55" alt="sum_(n=0)^(infty)((-1)^n(28n+3)(1/2)_n(1/4)_n(3/4)_n)/((n!)^33^n4^(2n+1))" /></td><td align="right" width="10"> <div id="eqn108" class="eqnum"> (108) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline284.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="39" alt="4/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline285.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline286.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="247" height="55" alt="sum_(n=0)^(infty)((-1)^n(260n+23)(1/2)_n(1/4)_n(3/4)_n)/((n!)^3(18)^(2n+1))" /></td><td align="right" width="10"> <div id="eqn109" class="eqnum"> (109) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline287.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="44" height="46" alt="4/(pisqrt(5))" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline288.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline289.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="247" height="55" alt="sum_(n=0)^(infty)((-1)^n(644n+41)(1/2)_n(1/4)_n(3/4)_n)/((n!)^35^n(72)^(2n+1))" /></td><td align="right" width="10"> <div id="eqn110" class="eqnum"> (110) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline290.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="39" alt="4/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline291.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline292.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="283" height="55" alt="sum_(n=0)^(infty)((-1)^n(21460n+1123)(1/2)_n(1/4)_n(3/4)_n)/((n!)^3(882)^(2n+1))" /></td><td align="right" width="10"> <div id="eqn111" class="eqnum"> (111) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline293.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="45" height="44" alt="(2sqrt(3))/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline294.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline295.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="182" height="55" alt="sum_(n=0)^(infty)((8n+1)(1/2)_n(1/4)_n(3/4)_n)/((n!)^39^n)" /></td><td align="right" width="10"> <div id="eqn112" class="eqnum"> (112) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline296.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="56" height="46" alt="1/(2pisqrt(2))" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline297.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline298.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="190" height="55" alt="sum_(n=0)^(infty)((10n+1)(1/2)_n(1/4)_n(3/4)_n)/((n!)^39^(2n+1))" /></td><td align="right" width="10"> <div id="eqn113" class="eqnum"> (113) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline299.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="56" height="46" alt="1/(3pisqrt(3))" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline300.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline301.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="190" height="55" alt="sum_(n=0)^(infty)((40n+3)(1/2)_n(1/4)_n(3/4)_n)/((n!)^3(49)^(2n+1))" /></td><td align="right" width="10"> <div id="eqn114" class="eqnum"> (114) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline302.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="52" height="46" alt="2/(pisqrt(11))" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline303.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline304.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="207" height="55" alt="sum_(n=0)^(infty)((280n+19)(1/2)_n(1/4)_n(3/4)_n)/((n!)^3(99)^(2n+1))" /></td><td align="right" width="10"> <div id="eqn115" class="eqnum"> (115) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline305.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="56" height="46" alt="1/(2pisqrt(2))" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline306.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline307.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="247" height="55" alt="sum_(n=0)^(infty)((26390n+1103)(1/2)_n(1/4)_n(3/4)_n)/((n!)^3(99)^(4n+2))." /></td><td align="right" width="10"> <div id="eqn116" class="eqnum"> (116) </div> </td></tr> </table> </div> <p> These equations were first proved by Borwein and Borwein (1987a, pp. 177-187). Borwein and Borwein (1987b, 1988, 1993) proved other equations of this type, and Chudnovsky and Chudnovsky (1987) found similar equations for other transcendental constants (Bailey <i>et al. </i>2007, pp. 44-45). </p> <p> A complete list of independent known equations of this type is given by </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline308.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="39" alt="4/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline309.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline310.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="118" height="57" alt="sum_(n=0)^(infty)((6n+1)(1/2)_n^3)/(4^n(n!)^3)" /></td><td align="right" width="10"> <div id="eqn117" class="eqnum"> (117) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline311.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="23" height="39" alt="(16)/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline312.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline313.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="126" height="57" alt="sum_(n=0)^(infty)((42n+5)(1/2)_n^3)/(64^n(n!)^3)" /></td><td align="right" width="10"> <div id="eqn118" class="eqnum"> (118) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline314.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="23" height="39" alt="(32)/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline315.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline316.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="344" height="57" alt="sum_(n=0)^(infty)((42sqrt(5)n+5sqrt(5)+30n-1)(1/2)_n^3)/(64^n(n!)^3)((sqrt(5)-1)/2)^(8n)" /></td><td align="right" width="10"> <div id="eqn119" class="eqnum"> (119) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline317.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="31" height="42" alt="(5^(1/4))/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline318.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline319.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="396" height="58" alt="sum_(n=0)^(infty)((540sqrt(5)n-1200n-525+235sqrt(5))(1/2)_n^3(sqrt(5)-2)^(8n))/((n!)^3)" /></td><td align="right" width="10"> <div id="eqn120" class="eqnum"> (120) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline320.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="39" height="42" alt="(12^(1/4))/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline321.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline322.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="344" height="58" alt="sum_(n=0)^(infty)((24sqrt(3)n-36n-15+9sqrt(3))(1/2)_n^3(2-sqrt(3))^(4n))/((n!)^3)" /></td><td align="right" width="10"> <div id="eqn121" class="eqnum"> (121) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/PiFormulas/Inline323.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="43" height="21" alt="m=1" /> with nonalternating signs, </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline324.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="39" alt="2/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline325.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline326.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="376" height="58" alt="sum_(n=0)^(infty)((-1)^n(1/2)_n^3(12sqrt(2)n-12n-5+4sqrt(2))(sqrt(2)-1)^(4n))/((n!)^3)" /></td><td align="right" width="10"> <div id="eqn122" class="eqnum"> (122) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline327.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="39" alt="2/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline328.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline329.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="392" height="58" alt="sum_(n=0)^(infty)((-1)^n(1/2)_n^3(60n-24sqrt(5)n+23-10sqrt(5))(sqrt(5)-2)^(4n))/((n!)^3)" /></td><td align="right" width="10"> <div id="eqn123" class="eqnum"> (123) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline330.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="15" height="39" alt="2/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline331.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline332.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="339" height="57" alt="sum_(n=0)^(infty)((-1)^n(1/2)_n^3(420n-168sqrt(6)n+177-72sqrt(6)))/((n!)^3)" /></td><td align="right" width="10"> <div id="eqn124" class="eqnum"> (124) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline333.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="45" height="44" alt="(2sqrt(2))/pi" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline334.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline335.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="166" height="58" alt="sum_(n=0)^(infty)((-1)^n(1/2)_n^3(2sqrt(2))^(2n))/((n!)^3)" /></td><td align="right" width="10"> <div id="eqn125" class="eqnum"> (125) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/PiFormulas/Inline336.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="43" height="21" alt="m=1" /> with alternating signs, </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline337.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="32" height="42" alt="(128)/(pi^2)" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline338.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline339.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="244" height="57" alt="sum_(n=0)^(infty)((-1)^n(1/2)_n^5(820n^2+180n+13))/(32^(2n)(n!)^5)" /></td><td align="right" width="10"> <div id="eqn126" class="eqnum"> (126) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/PiFormulas/Inline340.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="23" height="42" alt="(32)/(pi^2)" /></td><td align="center" width="14"><img src="/images/equations/PiFormulas/Inline341.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/PiFormulas/Inline342.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="210" height="57" alt="sum_(n=0)^(infty)((-1)^n(1/2)_n^5(20n^2+8n+1))/(2^(2n)(n!)^5)" /></td><td align="right" width="10"> <div id="eqn127" class="eqnum"> (127) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/PiFormulas/Inline343.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="43" height="21" alt="m=2" /> (Guillera 2002, 2003, 2006), </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation42.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="283" height="57" alt=" (32)/(pi^3)=sum_(n=0)^infty((1/2)_n^7(168n^3+76n^2+14n+1))/(32^(2n)(n!)^5) " /></td><td align="right" width="3"> <div id="eqn128" class="eqnum"> (128) </div> </td></tr> </table> </div> <p> for <img src="/images/equations/PiFormulas/Inline344.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="43" height="21" alt="m=3" /> (Guillera 2002, 2003, 2006), and no others for <img src="/images/equations/PiFormulas/Inline345.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="41" height="21" alt="m>3" /> are known (Bailey <i>et al. </i>2007, pp. 45-48). </p> <p> Bellard gives the exotic formula </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation43.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="298" height="87" alt=" pi=1/(740025)[sum_(n=1)^infty(3P(n))/((7n; 2n)2^(n-1))-20379280], " /></td><td align="right" width="3"> <div id="eqn129" class="eqnum"> (129) </div> </td></tr> </table> </div> <p> where </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation44.svg" data-src-small="/images/equations/PiFormulas/NumberedEquation44_400.svg" data-src-default="/images/equations/PiFormulas/NumberedEquation44.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="753" height="20" data-big="753 20" data-small="346 69" border="0" alt=" P(n)=-885673181n^5+3125347237n^4-2942969225n^3+1031962795n^2-196882274n+10996648. " /></td><td align="right" width="3"> <div id="eqn130" class="eqnum"> (130) </div> </td></tr> </table> </div> <p> Gasper quotes the result </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation45.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="247" height="39" alt=" pi=(16)/3[lim_(x->infty)x_1F_2(1/2;2,3;-x^2)]^(-1), " /></td><td align="right" width="3"> <div id="eqn131" class="eqnum"> (131) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/PiFormulas/Inline346.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="_1F_2" /> is a <a href="/GeneralizedHypergeometricFunction.html">generalized hypergeometric function</a>, and transforms it to </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation46.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="213" height="31" alt=" pi=lim_(x->infty)4x_1F_2(1/2;3/2,3/2;-x^2). " /></td><td align="right" width="3"> <div id="eqn132" class="eqnum"> (132) </div> </td></tr> </table> </div> <p> A fascinating result due to Gosper is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation47.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="289" height="53" alt=" lim_(n->infty)product_(i=n)^(2n)pi/(2tan^(-1)i)=4^(1/pi)=1.554682275.... " /></td><td align="right" width="3"> <div id="eqn133" class="eqnum"> (133) </div> </td></tr> </table> </div> <p> <img src="/images/equations/PiFormulas/Inline347.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> satisfies the <a href="/Inequality.html">inequality</a> </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation48.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="175" height="42" alt=" (1+1/pi)^(pi+1) approx 3.14097<pi. " /></td><td align="right" width="3"> <div id="eqn134" class="eqnum"> (134) </div> </td></tr> </table> </div> <p> D. Terr (pers. comm.) noted the curious identity </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/PiFormulas/NumberedEquation49.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="262" height="20" alt=" (3,1,4)=(1,5,9)+(2,6,5) (mod 10) " /></td><td align="right" width="3"> <div id="eqn135" class="eqnum"> (135) </div> </td></tr> </table> </div> <p> involving the first 9 digits of pi. </p> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/BBPFormula.html">BBP Formula</a>, <a href="/Digit-ExtractionAlgorithm.html">Digit-Extraction Algorithm</a>, <a href="/Pi.html">Pi</a>, <a href="/PiApproximations.html">Pi Approximations</a>, <a href="/PiContinuedFraction.html">Pi Continued Fraction</a>, <a href="/PiDigits.html">Pi Digits</a>, <a href="/PiIterations.html">Pi Iterations</a>, <a href="/PiSquared.html">Pi Squared</a>, <a href="/SpigotAlgorithm.html">Spigot Algorithm</a>       <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li> <a target="_blank" href="http://www.wolframalpha.com/entities/knots/figure_eight_knot/1j/jd/ti/"> figure eight knot </a> </li> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=pi"> pi </a> </li> <li> <a target="_blank" href="http://www.wolframalpha.com/input/?i=Catalan%27s+constant"> Catalan's constant </a> </li> </ul> </div> </div>   <h2>References</h2><cite>Adamchik, V. and Wagon, S. "A Simple Formula for <img src="/images/equations/PiFormulas/Inline348.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" />." <i>Amer. Math. Monthly</i> <b>104</b>, 852-855, 1997.</cite><cite>Adamchik, V. and Wagon, S. "Pi: A 2000-Year Search Changes Direction." <a href="http://www-2.cs.cmu.edu/~adamchik/articles/pi.htm">http://www-2.cs.cmu.edu/~adamchik/articles/pi.htm</a>.</cite><cite>Backhouse, N. "Note 79.36. Pancake Functions and Approximations to <img src="/images/equations/PiFormulas/Inline349.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" />." <i>Math. Gaz.</i> <b>79</b>, 371-374, 1995.</cite><cite>Bailey, D. H. "Numerical Results on the Transcendence of Constants Involving <img src="/images/equations/PiFormulas/Inline350.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" />, <img src="/images/equations/PiFormulas/Inline351.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="e" />, and Euler's Constant." <i>Math. Comput.</i> <b>50</b>, 275-281, 1988a.</cite><cite>Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/156881271X/ref=nosim/ericstreasuretro">Experimental Mathematics in Action.</a></i> Wellesley, MA: A K Peters, 2007.</cite><cite>Bailey, D. H.; Borwein, P.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." <i>Math. Comput.</i> <b>66</b>, 903-913, 1997.</cite><cite>Beck, G. and Trott, M. "Calculating Pi: From Antiquity to Moderns Times." <a href="http://documents.wolfram.com/mathematica/Demos/Notebooks/CalculatingPi.html">http://documents.wolfram.com/mathematica/Demos/Notebooks/CalculatingPi.html</a>.</cite><cite>Beckmann, P. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0312381859/ref=nosim/ericstreasuretro">A History of Pi, 3rd ed.</a></i> New York: Dorset Press, 1989.</cite><cite>Beeler, M. <i>et al. </i>Item 140 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. <i>HAKMEM.</i> Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 69, Feb. 1972. <a href="http://www.inwap.com/pdp10/hbaker/hakmem/pi.html#item140">http://www.inwap.com/pdp10/hbaker/hakmem/pi.html#item140</a>.</cite><cite>Berndt, B. C. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387941096/ref=nosim/ericstreasuretro">Ramanujan's Notebooks, Part IV.</a></i> New York: Springer-Verlag, 1994.</cite><cite>Beukers, F. "A Rational Approximation to <img src="/images/equations/PiFormulas/Inline352.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" />." <i>Nieuw Arch. Wisk.</i> <b>5</b>, 372-379, 2000.</cite><cite>Blatner, D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0802713327/ref=nosim/ericstreasuretro">The Joy of Pi.</a></i> New York: Walker, 1997.</cite><cite>Boros, G. and Moll, V. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521796369/ref=nosim/ericstreasuretro">Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals.</a></i> Cambridge, England: Cambridge University Press, 2004.</cite><cite>Borwein, J. and Bailey, D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/1568812116/ref=nosim/ericstreasuretro">Mathematics by Experiment: Plausible Reasoning in the 21st Century.</a></i> Wellesley, MA: A K Peters, 2003.</cite><cite>Borwein, J.; Bailey, D.; and Girgensohn, R. <i><a href="http://www.amazon.com/exec/obidos/ASIN/1568811365/ref=nosim/ericstreasuretro">Experimentation in Mathematics: Computational Paths to Discovery.</a></i> Wellesley, MA: A K Peters, 2004.</cite><cite>Borwein, J. M. and Borwein, P. B. <i><a href="http://www.amazon.com/exec/obidos/ASIN/047131515X/ref=nosim/ericstreasuretro">Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.</a></i> New York: Wiley, 1987a.</cite><cite>Borwein, J. M. and Borwein, P. B. "Ramanujan's Rational and Algebraic Series for <img src="/images/equations/PiFormulas/Inline353.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="1/pi" />." <i>Indian J. Math.</i> <b>51</b>, 147-160, 1987b.</cite><cite>Borwein, J. M. and Borwein, P. B. "More Ramanujan-Type Series for <img src="/images/equations/PiFormulas/Inline354.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="1/pi" />." In <i><a href="http://www.amazon.com/exec/obidos/ASIN/012058560X/ref=nosim/ericstreasuretro">Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987</a></i> (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin). New York: Academic Press, pp. 359-374, 1988.</cite><cite>Borwein, J. M. and Borwein, P. B. "Class Number Three Ramanujan Type Series for <img src="/images/equations/PiFormulas/Inline355.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="21" alt="1/pi" />." <i>J. Comput. Appl. Math.</i> <b>46</b>, 281-290, 1993.</cite><cite>Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. "Ramanujan, Modular Equations, and Approximations to Pi, or How to Compute One Billion Digits of Pi." <i>Amer. Math. Monthly</i> <b>96</b>, 201-219, 1989.</cite><cite>Borwein, J. M.; Borwein, D.; and Galway, W. F. "Finding and Excluding <img src="/images/equations/PiFormulas/Inline356.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="b" />-ary Machin-Type Individual Digit Formulae." <i>Canad. J. Math.</i> <b>56</b>, 897-925, 2004.</cite><cite>Castellanos, D. "The Ubiquitous Pi. Part I." <i>Math. Mag.</i> <b>61</b>, 67-98, 1988a.</cite><cite>Castellanos, D. "The Ubiquitous Pi. Part II." <i>Math. 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"Collection of Series for <img src="/images/equations/PiFormulas/Inline357.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" />." <a href="http://numbers.computation.free.fr/Constants/Pi/piSeries.html">http://numbers.computation.free.fr/Constants/Pi/piSeries.html</a>.</cite><cite>Guillera, J. "Some Binomial Series Obtained by the WZ-Method." <i>Adv. Appl. Math.</i> <b>29</b>, 599-603, 2002.</cite><cite>Guillera, J. "About a New Kind of Ramanujan-Type Series." <i>Exp. Math.</i> <b>12</b>, 507-510, 2003.</cite><cite>Guillera, J. "Generators of Some Ramanujan Formulas." <i>Ramanujan J.</i> <b>11</b>, 41-48, 2006.</cite><cite>Hardy, G. H. "A Chapter from Ramanujan's Note-Book." <i>Proc. Cambridge Philos. Soc.</i> <b>21</b>, 492-503, 1923.</cite><cite>Hardy, G. H. "Some Formulae of Ramanujan." <i>Proc. London Math. Soc.</i> (Records of Proceedings at Meetings) <b>22</b>, xii-xiii, 1924.</cite><cite>Hardy, G. H. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0821820230/ref=nosim/ericstreasuretro">Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed.</a></i> New York: Chelsea, 1999.</cite><cite>Le Lionnais, F. <i><a href="http://www.amazon.com/exec/obidos/ASIN/2705614079/ref=nosim/ericstreasuretro">Les nombres remarquables.</a></i> Paris: Hermann, 1983.</cite><cite>Lucas, S. K. "Integral Proofs that <img src="/images/equations/PiFormulas/Inline358.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="93" height="21" alt="355/113>pi" />." <i>Austral. Math. Soc. Gaz.</i> <b>32</b>, 263-266, 2005.</cite><cite>MathPages. "Rounding Up to Pi." <a href="http://www.mathpages.com/home/kmath001.htm">http://www.mathpages.com/home/kmath001.htm</a>.</cite><cite>Plouffe, S. "Identities Inspired from Ramanujan Notebooks (Part 2)." 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"A Spigot Algorithm for the Digits of <img src="/images/equations/PiFormulas/Inline362.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" />." <i>Amer. Math. Monthly</i> <b>102</b>, 195-203, 1995.</cite><cite>Ramanujan, S. "Modular Equations and Approximations to <img src="/images/equations/PiFormulas/Inline363.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" />." <i>Quart. J. Pure. Appl. Math.</i> <b>45</b>, 350-372, 1913-1914.</cite><cite>Sloane, N. J. A. Sequences <a href="http://oeis.org/A054387">A054387</a> and <a href="http://oeis.org/A054388">A054388</a> in "The On-Line Encyclopedia of Integer Sequences."</cite><cite>Smith, D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0486204294/ref=nosim/ericstreasuretro">History of Mathematics, Vol. 2.</a></i> New York: Dover, 1953.</cite><cite>Sondow, J. "Problem 88." <i>Math Horizons</i>, pp. 32 and 34, Sept. 1997.</cite><cite>Sondow, J. "A Faster Product for <img src="/images/equations/PiFormulas/Inline364.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="pi" /> and a New Integral for <img src="/images/equations/PiFormulas/Inline365.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="57" height="21" alt="ln(pi/2)" />." <i>Amer. Math. Monthly</i> <b>112</b>, 729-734, 2005.</cite><cite>Vardi, I. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0685479412/ref=nosim/ericstreasuretro">Computational Recreations in Mathematica.</a></i> Reading, MA: Addison-Wesley, p. 159, 1991.</cite><cite>Vieta, F. <i>Uriorum de rebus mathematicis responsorum.</i> Liber VII. 1593. Reprinted in New York: Georg Olms, pp. 398-400 and 436-446, 1970.</cite><cite>Wells, D. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0140080295/ref=nosim/ericstreasuretro">The Penguin Dictionary of Curious and Interesting Numbers.</a></i> Middlesex, England: Penguin Books, 1986.</cite><cite>Wolfram Research, Inc. "Some Notes On Internal Implementation: Mathematical Constants." <a href="http://reference.wolfram.com/language/tutorial/SomeNotesOnInternalImplementation.html">http://reference.wolfram.com/language/tutorial/SomeNotesOnInternalImplementation.html</a>.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/pi_formulas/0d/9o/pb/" title="Pi Formulas" target="_blank">Pi Formulas</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Pi Formulas." 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