One can find the Maclaurin series for by naïvely integrating term-by-term:
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 can instead be written as the finite sum,[4]
Again integrating both sides,
In the limit as the integral on the right above tends to zero when because
Therefore,
Convergence
The series for and converge within the complex disk , where both functions are holomorphic. They diverge for because when , there is a pole:
When the partial sums alternate between the values and never converging to the value
However, its term-by-term integral, the series for (barely) converges when because disagrees with its series only at the point so the difference in integrals can be made arbitrarily small by taking sufficiently many terms:
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 Finding ways to get around this slow convergence has been a subject of great mathematical interest.
^ Boyer, Carl B.; Merzbach, Uta C. (1989) [1968]. A History of Mathematics (2nd ed.). Wiley. pp. 428–429. ISBN9780471097631.
^For example: Gupta 1973;Joseph, George Gheverghese (2011) [1st ed. 1991]. The Crest of the Peacock: Non-European Roots of Mathematics (3rd ed.). Princeton University Press. p. 428.Levrie, Paul (2011). "Lost and Found: An Unpublished ζ(2)-Proof". Mathematical Intelligencer. 33: 29–32. doi:10.1007/s00283-010-9179-y. S2CID 121133743.
^ Shirali, Shailesh A. (1997). "Nīlakaṇṭha, Euler and π". Resonance. 2 (5): 29–43. doi:10.1007/BF02838013. S2CID 121433151. Also see the erratum: Shirali, Shailesh A. (1997). "Addendum to 'Nīlakaṇṭha, Euler and π'". Resonance. 2 (11): 112. doi:10.1007/BF02862651.
^K.V. Sarma (ed.). (PDF) (in Sanskrit and English). Translated by V.S. Narasimhan. Indian National Academy of Science. p. 48. Archived from the original (PDF) on 9 March 2012. Retrieved 17 January 2010.
^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
^K. V. Sarma & S Hariharan (ed.). (PDF). Yuktibhāṣā of Jyeṣṭhadeva. Archived from the original (PDF) on 28 September 2006. Retrieved 2006-07-09.
^C.K. Raju (2007). Cultural Foundations of Mathematics : Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16 c. CE. History of Science, Philosophy and Culture in Indian Civilisation. Vol. X Part 4. New Delhi: Centre for Studies in Civilisation. p. 231. ISBN978-81-317-0871-2.
Gupta, Radha Charan (1973). "The Madhava–Gregory series". The Mathematics Education. 7: B67–B70.
Horvath, Miklos (1983). "On the Leibnizian quadrature of the circle" (PDF). Annales Universitatis Scientiarum Budapestiensis (Sectio Computatorica). 4: 75–83.
Roy, Ranjan (1990). "The Discovery of the Series Formula for π by Leibniz, Gregory and Nilakantha" (PDF). Mathematics Magazine. 63 (5): 291–306. doi:10.1080/0025570X.1990.11977541.
April 14, 2023
gregory, series, mathematics, inverse, tangent, function, infinite, taylor, series, expansion, origin, arctan, displaystyle, arctan, frac, frac, frac, cdots, infty, frac, this, series, converges, complex, disk, displaystyle, except, displaystyle, where, arctan. In mathematics Gregory s series for the inverse tangent function is its infinite Taylor series expansion at the origin 1 arctan x x x 3 3 x 5 5 x 7 7 k 0 1 k x 2 k 1 2 k 1 displaystyle arctan x x frac x 3 3 frac x 5 5 frac x 7 7 cdots sum k 0 infty frac 1 k x 2k 1 2k 1 This series converges in the complex disk x 1 displaystyle x leq 1 except for x i displaystyle x pm i where arctan i displaystyle arctan pm i infty It was first discovered in the 14th century by Madhava of Sangamagrama c 1340 c 1425 as credited by Madhava s Kerala school follower Jyeṣṭhadeva s Yuktibhaṣa c 1530 In recent literature it is sometimes called the Madhava Gregory series to recognize Madhava s priority see also Madhava series 2 It was independently rediscovered by James Gregory in 1671 and by Gottfried Leibniz in 1673 who obtained the Leibniz formula for p as the special case 3 p 4 arctan 1 1 1 3 1 5 1 7 displaystyle frac pi 4 arctan 1 1 frac 1 3 frac 1 5 frac 1 7 cdots Contents 1 Proof 2 Convergence 3 History 4 See also 5 Notes 6 ReferencesProof Edit The derivative of arctan x is 1 1 x2 conversely the integral of 1 1 x2 is arctan x If y arctan x displaystyle y arctan x then tan y x displaystyle tan y x The derivative is d x d y sec 2 y 1 tan 2 y displaystyle frac dx dy sec 2 y 1 tan 2 y Taking the reciprocal d y d x 1 1 tan 2 y 1 1 x 2 displaystyle frac dy dx frac 1 1 tan 2 y frac 1 1 x 2 This sometimes is used as a definition of the arctangent arctan x 0 x d u 1 u 2 displaystyle arctan x int 0 x frac du 1 u 2 The Maclaurin series for x arctan x 1 1 x 2 textstyle x mapsto arctan x 1 big left 1 x 2 right is a geometric series 1 1 x 2 1 x 2 x 4 x 6 k 0 x 2 k displaystyle frac 1 1 x 2 1 x 2 x 4 x 6 cdots sum k 0 infty bigl x 2 bigr vphantom k One can find the Maclaurin series for arctan displaystyle arctan by naively integrating term by term 0 x d u 1 u 2 0 x 1 u 2 u 4 u 6 d u x 1 3 x 2 1 5 x 5 1 7 x 7 k 0 1 k x 2 k 1 2 k 1 displaystyle begin aligned int 0 x frac du 1 u 2 amp int 0 x left 1 u 2 u 4 u 6 cdots right du 5mu amp x frac 1 3 x 2 frac 1 5 x 5 frac 1 7 x 7 cdots sum k 0 infty frac 1 k x 2k 1 2k 1 end aligned 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 1 displaystyle x leq 1 arctan displaystyle arctan can instead be written as the finite sum 4 1 1 x 2 1 x 2 x 4 x 2 N x 2 N 1 1 x 2 displaystyle frac 1 1 x 2 1 x 2 x 4 cdots bigl x 2 bigr vphantom N frac bigl x 2 bigr N 1 1 x 2 Again integrating both sides 0 x d u 1 u 2 k 0 N 1 k x 2 k 1 2 k 1 0 x u 2 N 1 1 u 2 d u displaystyle int 0 x frac du 1 u 2 sum k 0 N frac 1 k x 2k 1 2k 1 int 0 x frac bigl u 2 bigr N 1 1 u 2 du In the limit as N displaystyle N to infty the integral on the right above tends to zero when x 1 displaystyle x leq 1 because 0 x u 2 N 1 1 u 2 d u 0 1 u 2 N 2 1 u 2 d u lt 0 1 u 2 N 2 d u 1 2 N 3 0 displaystyle begin aligned Biggl int 0 x frac bigl u 2 bigr N 1 1 u 2 du Biggr amp leq int 0 1 frac u 2N 2 1 u 2 du 5mu amp lt int 0 1 u 2N 2 du frac 1 2N 3 to 0 end aligned Therefore arctan x k 0 1 k x 2 k 1 2 k 1 displaystyle begin aligned arctan x sum k 0 infty frac 1 k x 2k 1 2k 1 end aligned Convergence EditThe series for arctan textstyle arctan and arctan displaystyle arctan converge within the complex disk x lt 1 displaystyle x lt 1 where both functions are holomorphic They diverge for x gt 1 displaystyle x gt 1 because when x i displaystyle x pm i there is a pole 1 1 i 2 1 1 1 1 0 displaystyle frac 1 1 i 2 frac 1 1 1 frac 1 0 infty When x 1 displaystyle x pm 1 the partial sums k 0 n x 2 k textstyle sum k 0 n x 2 k alternate between the values 0 displaystyle 0 and 1 displaystyle 1 never converging to the value arctan 1 1 2 textstyle arctan pm 1 tfrac 1 2 However its term by term integral the series for arctan textstyle arctan barely converges when x 1 displaystyle x pm 1 because arctan displaystyle arctan disagrees with its series only at the point 1 displaystyle pm 1 so the difference in integrals can be made arbitrarily small by taking sufficiently many terms lim N 0 1 1 1 u 2 k 0 N u 2 k d u 0 displaystyle lim N to infty int 0 1 biggl frac 1 1 u 2 sum k 0 N u 2 k biggr 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 1 4 p textstyle tfrac 1 4 pi Finding ways to get around this slow convergence has been a subject of great mathematical interest History EditThe earliest person to whom the series can be attributed with confidence is Madhava of Sangamagrama c 1340 c 1425 The original reference as with much of Madhava 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 displaystyle arctan include Nilakantha Somayaji s Tantrasangraha c 1500 5 6 Jyeṣṭhadeva s Yuktibhaṣa c 1530 7 and the Yukti dipika commentary by Sankara Variyar where it is given in verses 2 206 2 209 8 See also EditList of mathematical series Madhava seriesNotes Edit Boyer Carl B Merzbach Uta C 1989 1968 A History of Mathematics 2nd ed Wiley pp 428 429 ISBN 9780471097631 For example Gupta 1973 Joseph George Gheverghese 2011 1st ed 1991 The Crest of the Peacock Non European Roots of Mathematics 3rd ed Princeton University Press p 428 Levrie Paul 2011 Lost and Found An Unpublished z 2 Proof Mathematical Intelligencer 33 29 32 doi 10 1007 s00283 010 9179 y S2CID 121133743 Roy 1990 Shirali Shailesh A 1997 Nilakaṇṭha Euler and p Resonance 2 5 29 43 doi 10 1007 BF02838013 S2CID 121433151 Also see the erratum Shirali Shailesh A 1997 Addendum to Nilakaṇṭha Euler and p Resonance 2 11 112 doi 10 1007 BF02862651 K V Sarma ed Tantrasamgraha with English translation PDF in Sanskrit and English Translated by V S Narasimhan Indian National Academy of Science p 48 Archived from the original PDF on 9 March 2012 Retrieved 17 January 2010 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 K V Sarma amp S Hariharan ed A book on rationales in Indian Mathematics and Astronomy An analytic appraisal PDF Yuktibhaṣa of Jyeṣṭhadeva Archived from the original PDF on 28 September 2006 Retrieved 2006 07 09 C K Raju 2007 Cultural Foundations of Mathematics Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16 c CE History of Science Philosophy and Culture in Indian Civilisation Vol X Part 4 New Delhi Centre for Studies in Civilisation p 231 ISBN 978 81 317 0871 2 References EditBerggren Lennart Borwein Jonathan Borwein Peter eds 2004 Pi A Source Book 3rd ed Springer doi 10 1007 978 1 4757 4217 6 ISBN 978 1 4419 1915 1 Gupta Radha Charan 1973 The Madhava Gregory series The Mathematics Education 7 B67 B70 Horvath Miklos 1983 On the Leibnizian quadrature of the circle PDF Annales Universitatis Scientiarum Budapestiensis Sectio Computatorica 4 75 83 Roy Ranjan 1990 The Discovery of the Series Formula for p by Leibniz Gregory and Nilakantha PDF Mathematics Magazine 63 5 291 306 doi 10 1080 0025570X 1990 11977541 Retrieved from https en wikipedia org w index php title Gregory 27s series amp oldid 1144729052, wikipedia, wiki, book, books, library,