section_path: History/Infinite series
section_name: Infinite series
paragraph_text: Infinite series
  
  The calculation of π was revolutionized by the development of infinite series techniques in the 16th and 17th centuries. An infinite series is the sum of the terms of an infinite sequence. Infinite series allowed mathematicians to compute π with much greater precision than Archimedes and others who used geometrical techniques. Although infinite series were exploited for π most notably by European mathematicians such as James Gregory and Gottfried Wilhelm Leibniz, the approach also appeared in the Kerala school sometime in the 14th or 15th century. Around 1500 AD, a written description of an infinite series that could be used to compute π was laid out in Sanskrit verse in Tantrasamgraha by Nilakantha Somayaji. The series are presented without proof, but proofs are presented in a later work, Yuktibhāṣā, from around 1530 AD. Several infinite series are described, including series for sine (which Nilakantha attributes to Madhava of Sangamagrama), cosine, and arctangent which are now sometimes referred to as Madhava series. The series for arctangent is sometimes called Gregory's series or the Gregory–Leibniz series. Madhava used infinite series to estimate π to 11 digits around 1400.
  
  In 1593, François Viète published what is now known as Viète's formula, an infinite product (rather than an infinite sum, which is more typically used in π calculations): 2 π = 2 2 ⋅ 2 + 2 2 ⋅ 2 + 2 + 2 2 ⋯ {\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }
  
  In 1655, John Wallis published what is now known as Wallis product, also an infinite product: π 2 = (2 1 ⋅ 2 3) ⋅ (4 3 ⋅ 4 5) ⋅ (6 5 ⋅ 6 7) ⋅ (8 7 ⋅ 8 9) ⋯ {\displaystyle {\frac {\pi }{2}}={\Big (}{\frac {2}{1}}\cdot {\frac {2}{3}}{\Big)}\cdot {\Big (}{\frac {4}{3}}\cdot {\frac {4}{5}}{\Big)}\cdot {\Big (}{\frac {6}{5}}\cdot {\frac {6}{7}}{\Big)}\cdot {\Big (}{\frac {8}{7}}\cdot {\frac {8}{9}}{\Big)}\cdots }
  
  In the 1660s, the English scientist Isaac Newton and German mathematician Gottfried Wilhelm Leibniz discovered calculus, which led to the development of many infinite series for approximating π. Newton himself used an arcsine series to compute a 15-digit approximation of π in 1665 or 1666, writing, "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."
  
  In 1671, James Gregory, and independently, Leibniz in 1673, discovered the Taylor series expansion for arctangent: arctan z = z − z 3 3 + z 5 5 − z 7 7 + ⋯ {\displaystyle \arctan z=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots }
  
  This series, sometimes called the Gregory–Leibniz series, equals π 4 {\textstyle {\frac {\pi }{4}}} when evaluated with z = 1 {\displaystyle z=1}. But for z = 1 {\displaystyle z=1}, it converges impractically slowly (that is, approaches the answer very gradually), taking about ten times as many terms to calculate each additional digit.
  
  In 1699, English mathematician Abraham Sharp used the Gregory–Leibniz series for z = 1 3 {\textstyle z={\frac {1}{\sqrt {3}}}} to compute π to 71 digits, breaking the previous record of 39 digits, which was set with a polygonal algorithm.
  
  In 1706, John Machin used the Gregory–Leibniz series to produce an algorithm that converged much faster: π 4 = 4 arctan 1 5 − arctan 1 239. {\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}.}
  
  Machin reached 100 digits of π with this formula. Other mathematicians created variants, now known as Machin-like formulae, that were used to set several successive records for calculating digits of π.
  
  Isaac Newton accelerated the convergence of the Gregory–Leibniz series in 1684 (in an unpublished work; others independently discovered the result):
  
  Leonhard Euler popularized this series in his 1755 differential calculus textbook, and later used it with Machin-like formulae, including π 4 = 5 arctan 1 7 + 2 arctan 3 79, {\textstyle {\tfrac {\pi }{4}}=5\arctan {\tfrac {1}{7}}+2\arctan {\tfrac {3}{79}},} with which he computed 20 digits of π in one hour.
  
  Machin-like formulae remained the best-known method for calculating π well into the age of computers, and were used to set records for 250 years, culminating in a 620-digit approximation in 1946 by Daniel Ferguson – the best approximation achieved without the aid of a calculating device.
  
  In 1844, a record was set by Zacharias Dase, who employed a Machin-like formula to calculate 200 decimals of π in his head at the behest of German mathematician Carl Friedrich Gauss.
  
  In 1853, British mathematician William Shanks calculated π to 607 digits, but made a mistake in the 528th digit, rendering all subsequent digits incorrect. Though he calculated an additional 100 digits in 1873, bringing the total up to 707, his previous mistake rendered all the new digits incorrect as well.