Arctangent series explained

In mathematics, the arctangent series, traditionally called Gregory's series, is the Taylor series expansion at the origin of the arctangent function:

\arctan x = x - (x ^{3})/(3) + (x ^{5})/(5) - (x ^{7})/(7) + … = \sum ^{inf ty} _{k =0} (( -1 ) ^{kx} ^{2k+1})/(2 k +1) .

This series converges in the complex disk

|x| \leq 1 , except for x = \pm i (where

It was first discovered in the 14th century by Indian mathematician Mādhava of Sangamagrāma (– c. 1425), the founder of the Kerala school, and is described in extant works by Nīlakaṇṭha Somayāji (c. 1500) and Jyeṣṭhadeva (c. 1530). Mādhava's work was unknown in Europe, and the arctangent series was independently rediscovered by James Gregory in 1671 and by Gottfried Leibniz in 1673. In recent literature the arctangent series is sometimes called the Mādhava–Gregory series to recognize Mādhava's priority (see also Mādhava series).^[1]

The special case of the arctangent of is traditionally called the Leibniz formula for , or recently sometimes the Mādhava–Leibniz formula:

(\pi 4)/(=) \arctan 1 = 1 - (1)/(3) + (1)/(5) - (1)/(7) + … .

The extremely slow convergence of the arctangent series for

|x| ≈ 1 makes this formula impractical per se. Kerala-school mathematicians used additional correction terms to speed convergence. John Machin (1706) expressed as a sum of arctangents of smaller values, eventually resulting in a variety of Machin-like formulas for . Isaac Newton (1684) and other mathematicians accelerated the convergence of the series via various transformations.

Proof

If

y = \arctan x then \tan y = x. The derivative is(dx)/(dy) = \sec ^{2} y = 1 + \tan ^{2} y.

Taking the reciprocal,

(dy)/(dx) = (1)/(1+ \tan ^{2} y) = (1)/(1+ x ^{2}) .

This sometimes is used as a definition of the arctangent:

\arctan x = \int ^{x (du)/(1+ u ^{2})} _{0} .

The Maclaurin series for $x\; \backslash mapsto\; \backslash arctan\text{'}\; x\; =\; 1\; \backslash big/\; \backslash left(1\; +\; x^2\backslash right)$ is a geometric series:

(1)/(1+ x ^{2}) = 1 - x ^{2} + x ^{4} - x ^{6} + … = \sum ^{inf ty} _{k =0} l ({ - x ^{2}r){\vphantom)}} ^{k.}

One can find the Maclaurin series for

\arctan by naïvely integrating term-by-term:\begin{align} \int ^{x} _{0} (du)/(1+ u ^{2}) & = \int ^{x} _{0} \left( 1 - u ^{2} + u ^{4} - u ^{6} + … \right)du \\[ 5 mu] & = x - (13)/(x) ^{3} + (15)/(x) ^{5} - (17)/(x) ^{7} + … = \sum ^{inf ty} _{k =0} (( -1 ) ^{kx} ^{2k+1})/(2 k +1) . \end{align}

While this turns out correctly, integrals and infinite sums cannot always be exchanged in this manner. To prove that the integral on the left converges to the sum on the right for real

|x| \leq 1 , \arctan' can instead be written as the finite sum,^[2] (1)/(1+ x ^{2}) = 1 - x ^{2} + x ^{4} - … + l ({ - x ^{2}r){\vphantom)}} ^{N} + (l ({ - x ^{2})/(r ){}) ^{N+1} }.

Again integrating both sides,

\int ^{x} _{0} (du)/(1+ u ^{2}) = \sum ^{N} _{k =0} (( -1 ) ^{kx} ^{2k+1})/(2 k +1) + \int ^{x (l ({ - u ^{2})/(r ){})} _{0} ^{N+1} }\,du.

In the limit as

N \to inf ty, the integral on the right above tends to zero when |x| \leq 1 , because\begin{align} l | \int ^{x (l ({ - u ^{2})/(r ){})} _{0} ^{N+1} }\,du \,\Biggr|\,&\leq \int_0^1 \frac\,du\\[5mu]&< \int_0^1 u^du\,=\, \frac \,\to\, 0.\end

Therefore,

\begin{align} \arctan x = \sum ^{inf ty} _{k =0} (( -1 ) ^{kx} ^{2k+1})/(2 k +1) . \end{align}

Convergence

The series for $\backslash arctan\text{'}$ and

\arctan converge within the complex disk |x| < 1 , where both functions are holomorphic. They diverge for |x| > 1 because when x = \pm i , there is a pole:(1 { 1)/(+) i ^{2}} = (1 { 1)/(-) 1 } = (10)/(=) inf ty.

When

x = \pm 1 , the partial sums $\backslash sum\_^n\; (-x^2)^k$ alternate between the values 0 and 1 , never converging to the value $\backslash arctan\text{'}(\backslash pm1)\; =\; \backslash tfrac12.$

However, its term-by-term integral, the series for $\backslash arctan,$ (barely) converges when

x = \pm 1 , because \arctan' disagrees with its series only at the point \pm 1 , so the difference in integrals can be made arbitrarily small by taking sufficiently many terms: \lim _{N} \int ^{1 l ( (1)/(1+ u ^{2})} _{0} - \sum ^{N} _{k =0} ( - u ^{2)} ^{kr)du} = 0

Because of its exceedingly slow convergence (it takes five billion terms to obtain 10 correct decimal digits), the Leibniz formula is not a very effective practical method for computing $\backslash tfrac14\backslash pi.$ Finding ways to get around this slow convergence has been a subject of great mathematical interest.

Accelerated series

Isaac Newton accelerated the convergence of the arctangent series in 1684 (in an unpublished work; others independently discovered the result and it was later popularized by Leonhard Euler's 1755 textbook; Euler wrote two proofs in 1779), yielding a series converging for

\begin{align} \arctan x & = (x)/(1+ x ^{2}) \sum ^{inf ty} _{n =0} \prod ^{n} _{k =1} (2 k)/(2 k +1) (x ^{2})/(1+ x ^{2}) \\[ 10 mu] & = (x)/(1+ x ^{2}) + (23)/((x ^{3})/(( 1+ x ^{2)\vphantom{l})) ^{2}} + (2 ⋅ 4)/(3 ⋅ 5) (x ^{5})/(( 1+ x ^{2)\vphantom{l}) ^{3}} + (2 ⋅ 4 ⋅ 6)/(3 ⋅ 5 ⋅ 7) (x ^{7})/(( 1+ x ^{2)\vphantom{l}) ^{4}} + … \\[ 10 mu] & = C(x)\left( S(x) + (23 S(x))/(^{3}) + (2 ⋅ 4)/(3 ⋅ 5) S(x) ^{5} + (2 ⋅ 4 ⋅ 6)/(3 ⋅ 5 ⋅ 7) S(x) ^{7} + … \right), \end{align} where $\backslash vphantom\backslash Big|\; C(x)\; =\; 1\; \backslash big/\; \backslash !\backslash sqrt\; =$\cos(\arctan x) and $\backslash vphantom\backslash Big|\; S(x)\; =\; x\; \backslash big/\; \backslash !\backslash sqrt\; =$$\backslash sin(\backslash arctan\; x).$

Each term of this modified series is a rational function with its poles at

x = \pm i in the complex plane, the same place where the arctangent function has its poles. By contrast, a polynomial such as the Taylor series for arctangent forces all of its poles to infinity.

History

The earliest person to whom the series can be attributed with confidence is Mādhava of Sangamagrāma (c. 1340 – c. 1425). The original reference (as with much of Mādhava's work) is lost, but he is credited with the discovery by several of his successors in the Kerala school of astronomy and mathematics founded by him. Specific citations to the series for

\arctan include Nīlakaṇṭha Somayāji's Tantrasaṅgraha (c. 1500),^{[3] [4]} Jyeṣṭhadeva's Yuktibhāṣā (c. 1530),^[5] and the Yukti-dipika commentary by Sankara Variyar, where it is given in verses 2.206  - 2.209.^[6]

See also

• List of mathematical series
• Mādhava series

References

• Book: Berggren . Lennart . Borwein . Jonathan . Jonathan Borwein . Borwein . Peter . Peter Borwein . 2004 . Pi: A Source Book . 3rd . Springer . 10.1007/978-1-4757-4217-6 . 978-1-4419-1915-1 .
• Gupta . Radha Charan . Radha Charan Gupta . 1973 . The Mādhava–Gregory series . The Mathematics Education . 7 . B67–B70 .
  • Horvath . Miklos . 1983. On the Leibnizian quadrature of the circle. . Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica) . 4 . 75–83 .
• Roy . Ranjan . 1990 . The Discovery of the Series Formula for by Leibniz, Gregory and Nilakantha . Mathematics Magazine . 63 . 5 . 291–306 . 10.1080/0025570X.1990.11977541.

Notes and References

1. For example:, ; Book: Joseph, George Gheverghese . George Gheverghese Joseph . The Crest of the Peacock: Non-European Roots of Mathematics . Princeton University Press . 2011 . 3rd . 428 . 1st ed. 1991. Levrie . Paul . 2011 . Lost and Found: An Unpublished -Proof . Mathematical Intelligencer . 33 . 29–32 . 10.1007/s00283-010-9179-y . 121133743. Other combinations of names include, Madhava–Gregory–Leibniz series: Benko . David . Molokach . John . 2013 . The Basel Problem as a Rearrangement of Series . College Mathematics Journal . 44 . 3 . 171–176 . 10.4169/college.math.j.44.3.171 . 124737638 . Madhava–Leibniz–Gregory series: Book: Danesi, Marcel . 2021 . 1. Discovery of π and Its Manifestations . Pi in Nature, Art, and Culture . Brill . 1–30 . 10.1163/9789004433397_002 . 978-90-04-43337-3 . 242107102 . Nilakantha–Gregory series: Campbell . Paul J. . 2004 . Reviews . Borwein, Jonathan, and David Bailey, Mathematics by Experiment . Mathematics Magazine . 77 . 2 . 163 . 10.1080/0025570X.2004.11953245. 218541218 . Gregory–Leibniz–Nilakantha formula: Gawrońska . Natalia . Słota . Damian . Wituła . Roman . Zielonka . Adam . 2013 . Some generalizations of Gregory's power series and their applications . Journal of Applied Mathematics and Computational Mechanics . 12 . 3 . 79–91 . 10.17512/jamcm.2013.3.09 .
2. Shirali . Shailesh A. . 1997 . Nīlakaṇṭha, Euler and . Resonance . 2 . 5 . 29–43 . 10.1007/BF02838013 . 121433151 . Also see the erratum: Shirali . Shailesh A. . 1997 . Addendum to 'Nīlakaṇṭha, Euler and ' . Resonance . 2 . 11 . 112 . 10.1007/BF02862651 . free .
3. Web site: Tantrasamgraha with English translation . K.V. Sarma . Translated by V.S. Narasimhan . Indian National Academy of Science . 48 . Sanskrit, English . 17 January 2010 . dead . https://web.archive.org/web/20120309014402/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_2/20005a5d_s1.pdf . 9 March 2012 .
4. Tantrasamgraha, ed. K.V. Sarma, trans. V. S. Narasimhan in the Indian Journal of History of Science, issue starting Vol. 33, No. 1 of March 1998
5. Web site: K. V. Sarma & S Hariharan. Yuktibhāṣā of Jyeṣṭhadeva. A book on rationales in Indian Mathematics and Astronomy—An analytic appraisal. 2006-07-09. https://web.archive.org/web/20060928203221/http://www.new.dli.ernet.in/insa/INSA_1/20005ac0_185.pdf . 28 September 2006.
6. Book: C.K. Raju. Cultural Foundations of Mathematics : Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16 c. CE. Centre for Studies in Civilisation. New Delhi. 2007. History of Science, Philosophy and Culture in Indian Civilisation. X Part 4. 231. 978-81-317-0871-2.