International Journal of Mathematics Research. ISSN 0976-5840 Volume 9, Number 1 (2017), pp. 1-4 © International Research Publication House http://www.irphouse.com (−1)푛−1 Approximation of the Series ∑∞ 푛=1 푎푛2+푏푛+푐 where a,b,c ∈ R with a≠0 Kumari Sreeja S. Nair and Dr. V. Madhukar Mallayya 1Assistant Professor, Department of Mathematics Govt. Arts College, Thiruvananthapuram, Kerala, India. 2Former Professor and Head, Department of Mathematics Mar Ivanios College, Thiruvananthapuram, Kerala, India. Abstract Here we give approximation of an alternating series using remainder term of the series. Here we introduce a new term called correction term. The correction term plays a vital role in series approximation. Keywords: Correction function, error function, remainder term, alternating series, rational approximation, Dirichlet’s series. INTRODUCTION The illusturious mathematician Madhava of 14th century introduces correction function for the series for pi. The Madhava series is 4푑 4푑 4푑 ..................... 푛−1 4푑 푛 4푑(2푛)/2 C = − + − +(−1) + (−1) , where C is the 1 3 5 2푛−1 (2푛)2+1 circumference of a circle of diameter d. n (2푛)/2 Here the remainder term is (-1) 4d Gn where Gn = is the correction (2푛)2+1 term. The introduction of the correction term improves the value of C and gives a better approximation for it. 2 Kumari Sreeja S. Nair and Dr. V. Madhukar Mallayya (−1)푛−1 RATIONAL APPROXIMATION OF ALTERNATING SERIES ∑∞ 푛=1 푎푛2+푏푛+푐 where a,b,c ∈ R with a≠0 and √푏2 − 4푎푐 ≠ 2a. (−1)푛−1 The alternating series ∑∞ satisfies the conditions of alternating series 푛=1 푎푛2+푏푛+푐 test and so it is convergent. Theorem (−1)푛−1 The correction function for the alternating series ∑∞ where a,b,c ∈ R with 푛=1 푎푛2+푏푛+푐 1 a≠ 0 is Gn = {2푎푛2+(2푏+2푎)푛+(2푐+푏+2푎)} Proof If Gn is the correction function after n terms of the series ,then 1 we have Gn + Gn+1 = 푎푛2+(2푎+푏)푛+푎+푏+푐 1 The error function is En = Gn + Gn+1 − 푎푛2+(2푎+푏)푛+푎+푏+푐 1 Let Gn (푟1 , 푟2) = 2 where 푟1 , 푟2 ∈ R and {2푎푛 +(4푎+2푏)푛+(2푎+2푏+2푐)}−(푟1 푛+푟2) n is fixed. Then error function |퐸푛(푟1, 푟2)| is minimum for r1 = 2a , r2 = b Hence for r1 = 2a , r2 = b , both Gn and En are functions of a single variable n. 푛−1 ∞ (−1) Thus the correction function for the series ∑ is 푛=1 푎푛2+푏푛+푐 1 Gn = {2푎푛2+(2푏+2푎)푛+(2푐+푏+2푎)} The corresponding error function is |(푏2−4푎푐)−4푎2| |En | = {2푎푛2+(2푏+2푎)푛+(2푐+푏+2푎)}{(2푎푛2+(2푏+6푎)푛+(6푎+3푏+2푐)}{(푎푛2+(2푎+푏)푛+(푎+푏+푐)} Hence the proof. Approximation of the Series 3 REMARK th Clearly Gn is less than the absolute value of the (n+1) term. APPLICATION 풏−ퟏ ∞ (−ퟏ) 1. The series ∑ = ᶯ(ퟐ) 풏=ퟏ 풏ퟐ We have ᶯ(2) = 0.8224670334, using a calculator. 1 The correction function for the series is Gn = 2푛2+2푛+2 For n= 10 , the series approximation after applying correction function is given below 푛 Number of terms Sn Sn + (−1) Gn 10 0.8179621756 0.82246666801 (−1)푛−1 2. THE ALTERNATING SERIES ∑∞ 푛=1 푛(푛+1) (−1)푛−1 The alternating series ∑∞ is convergent and converges to 2log2-1. 푛=1 푛(푛+1) We have 2log2-1 = 0.3862943611, using a calculator. 1 The correction function for the series is Gn = 2(푛+1)2+12 For n= 10 , the series approximation after applying correction function is given below 푛 Number of terms Sn Sn + (−1) Gn 10 0.3821789321 0.3863283098 CONCLUSION The introduction of correction function improves the sum of the series and gives a better approximation. 4 Kumari Sreeja S. Nair and Dr. V. Madhukar Mallayya 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 .