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Rational Approximation of the Series ∞ (−1)푛−1 ∑ 푝(푝 + 3) 푛=1

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Rational Approximation of the Series ∞ (−1)푛−1 ∑ 푝(푝 + 3) 푛=1 International Journal of Pure and Applied Mathematical Sciences. ISSN 0972-9828 Volume 10, Number 1 (2017), pp. 1-5 © Research India Publications http://www.ripublication.com Rational Approximation of the Series ∞ (−1)푛−1 ∑ 푝(푝 + 3) 푛=1 where p= 3n-2 Kumari Sreeja S. Nair1 and Dr. V. Madhukar Mallayya2 1Assistant Professor of Mathematics Govt. Arts College, Thiruvananthapuram, Kerala, India. 2Former Professor and Head, Department of Mathematics Mar Ivanios College Thiruvananthapuram, Kerala, India. Abstract In this paper we give approximation of an alternating series using remainder term of the series. Here we introduce a new term called correction term. The use of correction term gives a better approximation of series. Keywords: Correction function, error function, remainder term, alternating series, rational approximation. 2 Kumari Sreeja S. Nair and Dr. V. Madhukar Mallayya INTRODUCTION The illusturious mathematician Madhava of 14th century introduces correction for the series for pi. The Madhava series is 2푛 4푑 4푑 4푑 4푑 4푑( ) C = − + −..............± ∓ 2 , where + or –indicates that 1 3 5 2푛−1 (2푛)2+1 n is odd or even and C is the circumference of a circle of diameter d . or more specifically, 4푑 4푑 4푑 4푑 4푑(2푛)/2 C = − + −.....................+(−1)푛−1 + (−1)푛 1 3 5 2푛−1 (2푛)2+1 n (2푛)/2 Here the remainder term is (-1) 4d Gn where Gn = is the correction (2푛)2+1 function. The introduction of the correction term improves the value of C and gives a better approximation for it. RATIONAL APPROXIMATION OF ALTERNATING SERIES 푛−1 ∞ (−1) ∑ where p= 3n-2 푛=1 푝(푝+3) 푛−1 ∞ (−1) The alternating series ∑ where p= 3n-2 satisfies the conditions of 푛=1 푝(푝+3) alternating series test and so it is convergent. If Rn denotes the remainder term after n terms of the series, then 푛 Rn = (−1) Gn where Gn is the correction function after n terms of the series Theorem 1. 푛−1 ∞ (−1) The correction function for the alternating series ∑ where p= 3n-2 푛=1 푝(푝+3) 1 is Gn = 2(푝+3)2+32 Rational Approximation of the Series 3 Proof: If Gn denotes the correction function after n terms of the series, then it follows that 1 Gn + Gn+1 = where 푝 = 3푛 − 2 (푝+3)(푝+6) As n→n+1, we have p →(p+3) 1 The error function is En = Gn + Gn+1 − . (푝+3)(푝+6) 1 For a fixed n and for 푟1 , 푟2 ∈ R ,let Gn (푟1 , 푟2) = 2 2푝 + 18푝+36−(푟1푝+ 푟2) Then the error function is |퐸푛(푟1, 푟2)| is minimum for 푟1 = 푟2 =6 Thus for 푟1 = 푟2 = 6, both Gn and En are functions of a single variable n. (−1)푛−1 Thus the correction function for the series ∑∞ is 푛=1 푝(푝+3) 1 Gn = 2(푝+3)2+ 32 The corresponding error function is 92(1.3) |En | = {2(푝+3)2+32}{2(푝+6)2+32}{(푝+3)(푝+6)} Hence the proof. Remark : 1 th Clearly Gn < , absolute value of (n+1) term (푝+3)(푝+6) 4 Kumari Sreeja S. Nair and Dr. V. Madhukar Mallayya Theorem 2. 푛−1 ∞ (−1) The correction functions for series ∑ . where p= 2n-1 푛=1 푝 (푝+2) follow aninfinitcontinued fraction 1 42(1.3) 82(3.5) (2(푝+2)2+22)− (2(푝+2)2+62)− (2(푝+2)2+102)− 122(5.7) 162(7.9) (2(푝+2)2+142)− (2(푝+2)2+182)−⋯ Proof: 푛−1 ∞ (−1) 1 The correction function or the series ∑ is Gn = 푛=1 푝 (푝+2) 2(푝+2)2+ 22 where p= 2n-1. ퟏ The first order correction function is Gn(1) = ퟐ(풑+ퟐ)ퟐ+ ퟐퟐ 1 Now choose Gn(2) = 퐴 where 퐴1 and 푥 are any {2(푝+2)2+ 22}+ 1 {2(푝+2)2+ 22}+푥} two real numbers. Then error is minimum for 퐴1 = −48 and 푥= 32 ퟏ The second order correction function is Gn(2) = ퟒퟐ(ퟏ.ퟑ) {ퟐ(풑+ퟐ)ퟐ+ ퟐퟐ}− {ퟐ(풑+ퟐ)ퟐ+ ퟔퟐ} 1 Again choose Gn(3) = 2 2 2 4 (1.3) {2(푝+2) +2 }− 퐴 {2(푝+2)2+62}− 2 {2(푝+2)2+62}+푥 Then error is minimum for 퐴2 = −960 and 푥 =64 Rational Approximation of the Series 5 The third order correction function is ퟏ Gn(3) = ퟒퟐ(ퟏ.ퟑ) {ퟐ(풑+ퟐ)ퟐ+ퟐퟐ}− ퟖퟐ(ퟑ.ퟓ) {ퟐ(풑+ퟐ)ퟐ+ퟔퟐ}− {ퟐ(풑+ퟐ)ퟐ+ퟏퟎퟐ} Continuing this process we get the correction functions follow an infinite continued fraction pattern as follows 1 42(1.3) {2(푝+2)2+22}− 82(3.5) {2(푝+2)2+62}− 122(5.7) {2(푝+2)2+102}− 162(7.9) {2(푝+2)2+142}− {2(푝+2)2+182}−⋯ REFERENCES [1] Dr. Konrad Knopp - Theory and Application of Infinite series - Blackie and son limited (London and Glasgow) [2] Sankara and Narayana, Lilavati of Bhaskaracharya with the Kriyakramakari, an elaborate exposition of the rationale with introduction and appendices (ed) K.VSarma (Visvesvaranand Vedic Research Institute, Hoshiarpur) 1975, p, 386-391. [3] Dr.V.Madhukar Mallayya - Proceedings of the Conference on Recent Trends in Mathematical Analysis- © 2003,Allied Publishers Pvt.Ltd, ISBN 81-7764- 399-1 [4] A Course of Pure Mathematics -G.H.Hardy (tenth edition) Cambridge at the university press 1963 [5] K. Knopp, Infinite sequences .and series, Dover- 1956 [6] T.Hayashi, T.K. Kusuba and M.Yano,Centaururs,33,149,1990 [7] Yuktidipika of Sankara (commentary on Tantrasangraha), ed. K.V.Sarma, Hoshiarpur 1977 6 Kumari Sreeja S. Nair and Dr. V. Madhukar Mallayya .
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