-ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION

Size: px
Start display at page:

Download "-ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION"

Transcription

1 NEW NEWTON-TYPE METHOD WITH k -ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION R. Thukral Padé Research Cetre, 39 Deaswood Hill, Leeds West Yorkshire, LS7 JS, ENGLAND ABSTRACT The objective of this paper is to defie a ew Newto-type method for fidig simple root of a polyomial. It is proved that the ew oe-poit method has the covergece order of k requirig fuctio evaluatios per iteratio, where k is the umber of terms i the geeratig series. Kug ad Traub cojectured that the multipoit iteratio methods, without memory based o evaluatios, could achieve maximum covergece order, but the ew method produces covergece order of k, which is better tha the expected maximum covergece order of eight. Therefore, we show that the cojecture fails for a particular set of polyomial equatios. We will demostrate that the ew method is very simple to costruct. Keywords: Newto-type method; Polyomial equatio; Kug-Traub s cojecture; Efficiecy idex; Optimal order of covergece. Subject Classificatios: AMS (MOS): 6H0. INTRODUCTION I this paper, we preset a ew oe-poit k of a polyomial equatio. Let -order iterative method to fid a simple root p( x ) a a x a x a x a x be a polyomial of 3 q 0 3 degree q ad a i are real costats. Fidig zeros of a polyomial has bee iterest i theoretical ad may areas of applied mathematics. It is well established that may higher order multi-poit variats of the Newto-type method have bee developed based o the Kug ad Traub cojecture [4]. Here we preset a ew iterative method which has a better efficiecy idex tha the classical Newto method [3,,6,7,0]. This paper is actually a cotiuatio form the previous studies [8,9], hece the ew oe-poit method is applied to higher order polyomial equatio. For the purpose of this paper, we costruct a ew Newto-type iterative method of k - order for fidig simple root of polyomial equatios. The ew oe-poit method preseted i this paper oly uses evaluatios of the fuctio per iteratio. Kug ad Traub cojectured that the multipoit iteratio methods, without memory based o evaluatios, could achieve optimal covergece order. I fact, we have obtaied a higher order of covergece tha the maximum order of covergece suggested by Kug ad Traub cojecture [4]. We demostrate that the Kug ad Traub cojecture fails for a particular case. Progressive Academic Publishig, UK Page

2 Prelimiaries I order to establish the order of covergece of the iterative method [3,,6,0], some of the defiitios are stated: Defiitio Let f x be a real fuctio with a simple root ad let real umbers that coverge towards. The order of covergece p is give by x lim 0 p x x be a sequece of where p ad is the asymptotic error costat (AEC). Let e x be the error i the th iteratio, the the relatio p p e e e, () is the error equatio. If the error equatio exists, the p is the order of covergece of the iterative method, [3,,6,0]. Defiitio Let be the umber of fuctio evaluatios of the iterative method. The efficiecy of the iterative method is measured by the cocept of efficiecy idex ad defied as E, p p (3) where p is the order of covergece of the method, [6]. Defiitio 3 (Kug ad Traub cojecture) Let x g x () defie as a iterative fuctio without memory with -evaluatios. The pg p opt, (4) where p opt is the maximum order, [4]. Covergece Aalysis I this sectio we defie a ew class of oe-poit k -order method for fidig simple root of a polyomial equatio. I fact, the ew iterative method is a extesio of the Thukral s quadratic ad cubic methods, give i [8,9]. Hece, we will demostrate that the ew oe-poit method ca be costructed to fid a simple root of ay order of polyomial equatio. The order of covergece the ew iterative method is determied by the k, that is umber of terms i the geeratig series, hece depedig o k we ca costruct ay desired order of covergece. The Method The ew oe-poit k -order Newto-type method is expressed by x x rhk r, t, t, t3, tp () where k,,, 3,,,, 3, ; H r t t t t AEC r t t t t i r i k p p (6) i Progressive Academic Publishig, UK Page

3 f x f x f x r, t, t, f x! f x 3! f x f x f x f x t, t, t, iv v vi 3 4 4! f x! f x 6! f x vii viii ix f x f x f x t6, t7, t8, 7! f x 8! f x 9! f x where x is the iitial guess ad provided that deomiators of (7) are ot equal to zero. 0 Now, we shall verify the covergece property of the ew oe-poit k -order iterative method (). Theorem Let D be a simple zero of a sufficietly smooth fuctio f : D for a ope iterval D. If the iitial guess x 0 is sufficietly close to, the the covergece order of the ew oe-poit iterative method defied by () is k. Proof Let be a simple root of e x. f x, i.e. f 0 ad f 0 The Taylor series expasio ad takig ito accout f 0 (7), ad the error is expressed as, we have f ( x ) f e c e c e c e c e c e c e c e. (8) f ( x ) f c e 3c e 4c e c e 6c e 7c e 8c e. (9) f ( x ) f c 6c e c e 0c e 30c e 4c e 6c e. (0) f ( x ) f 6c 4c e c e 0c e 0c e 336c e () iv f ( x ) f 4c 0c e c e 840c e 680c e () v f ( x ) f 0c 70c e 3 0c e 670c e (3) vi f ( x ) f 70c 0c e 060c e (4) vii f ( x ) f 0c 4030c e. where () 7 8 Progressive Academic Publishig, UK Page 3

4 iv v vi f f f f f c, c3, c4, c, c6, f f f f f (6) Dividig (8) by (9), we have f x e 3 ce c c3 e. f x (7) ad f x c c 3c3 e c 4c 9c3 6 c4 e. fx (8) f x 6c3 c4 cc3 e. f x (9) iv f x 4c4 4c cc4 e. fx (0) iv f x 0c 403 c6 cc e. fx () vi f x 70c6 707c7 cc6 e. fx () vii f x 040c7 4030c cc7 e. fx (3) viii f x 4030c c cc8 e. fx (4) Substitutig (9) i (), we obtai f x e 3 e ce c c3 e, f x () Therefore, the asymptotic error costat give by () is expressed as AEC 0 ce. (6) Progressive Academic Publishig, UK Page 4

5 It is well kow that (6) is the asymptotic error costat for the classical Newto method. Therefore, we take error equatio (6) as our ext term of the geeratig series give i (). Furthermore, we obtai a family of higher iterative method by icreasig the terms of summatio series of (6). Hece, we show the asymptotic error costat AEC k for the k -order Newto-type method. I order to obtai the followig terms of the geeratig series, we must substitute the essetial parts i AEC k, thus c t, c3 t, c4 t3, c t4, c6 t, (7) e r. (8) where r ad t i are give by (7). We use this priciple to obtai the followig oe-poit k -order iterative method; Secod-order Polyomial equatio This method has bee recetly preseted i [8] ad we review the basic expasio,. k : Oe-poit third-order iterative method is give by k i x x r AEC i r x r tr i (9) ad the error equatio 3 AEC ce (30). k : Oe-poit fourth-order iterative method is give by x x r tr t r (3) ad the error equatio 3 4 AEC ce. (3) 3. k 3: Oe-poit fifth-order iterative method is give by 3 3 x x r tr t r t r (33) ad the error equatio 4 AEC 3 4 ce. (34) 4. k 4: Oe-poit sixth-order iterative method is give by x x r tr t r t r 4t r (3) ad the error equatio 6 AEC 4 4 ce. (36). k : Oe-poit seveth-order iterative method is give by Progressive Academic Publishig, UK Page

6 x x r tr t r t r 4t r 4t r (37) ad the error equatio 6 7 AEC 3 ce. (38) 6. k 6 : Oe-poit eighth-order iterative method is give by x x r tr t r t r 4t r 4t r 3t r (39) ad the error equatio 7 8 AEC 6 49 ce. (40) It is well established that the maximum order of covergece of optimal methods with three fuctios evaluatios is four. As illustrated above we have obtaied order of covergece greater tha four, hece the Kug ad Traub cojecture fails for k 3. To produce the ext oe-poit higher order of covergece with oly three fuctio evaluatios, we use the followig priciple. The process is very simple, usig the coefficiet of the error equatio AEC k as our coefficiet of the ext term of the geeratig series, thus we ca calculate the ext higher order of covergece method. Hece, k -order Newto-type method. AEC k is the error equatio of the Third-order Polyomial equatio This cubic equatio method has bee recetly preseted i [9] ad review some of the results of the method.. k : Oe-poit third-order iterative method is give by x x r tr (4) ad the error equatio AEC c 3 c3 e (4). k : Oe-poit fourth-order iterative method is give by x x r tr t t r (43) ad the error equatio AEC c 4 c c3 e (44) 3. k 3: Oe-poit fifth-order iterative method is give by 3 x x r tr t t r t t t r (4) ad the error equatio AEC 3 4c 4 3c3 c c3 e (46) 4. k 4: Oe-poit sixth-order iterative method is give by Progressive Academic Publishig, UK Page 6

7 3 4 4 x x r tr t t r t t t r 4t 3t t t r (47) ad the error equatio AEC 4 4c 4 6 3c c3 6c c3 e (48). k : Oe-poit seveth-order iterative method is give by x x r tr t t r t t t r 4t 3t t t r 4t 3t t 6t t r (49) ad the error equatio AEC 6c c3 30c c3 c c3 e (0) 6. k 6 : Oe-poit eighth-order iterative method is give by x x r tr t t r t t t r 4t 3t t t r t 3t t 6t t r 6t t 30t t t t r () ad the error equatio AEC 6 3c c c3 330c c3 49c c3 e () Due to Kug ad Traub cojecture the maximum order of covergece of optimal methods with four fuctios evaluatios is eight. As illustrated above whe k 7 we have obtaied order of covergece greater tha eight, hece the Kug ad Traub cojecture fails. Fourth-order Polyomial equatio. k : Oe-poit third-order iterative method is give by x x r tr (3) ad the error equatio AEC c 3 c3 e (4). k : Oe-poit fourth-order iterative method is give by x x r tr t t r () ad the error equatio AEC c 3 4 cc3 c4 e (6) 3. k 3: Oe-poit fifth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r (7) ad the error equatio AEC 3 4c 4 3c3 c c3 6cc4 e (8) 4. k 4: Oe-poit sixth-order iterative method is give by Progressive Academic Publishig, UK Page 7

8 x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 r (9) ad the error equatio AEC 4 4c 3 6 8c c3 84c c3 7c3c4 8c c4 e (60). k : Oe-poit seveth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r 4t t t t 6tt 3 r 4t 8tt 84t t 7tt3 8t t3 r (6) ad the error equatio AEC 3c c3 80c c3 330c c3 4c4 0c c4 7cc3 c4 e (6) 6. k 6 : Oe-poit eighth-order iterative method is give by x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 r t t t t 3 t t t t t r t 6 t 3 t t t 4 t t t 3 t t t t r 6 (63) ad the error equatio AEC 6 49c c c3 990c c3 87c c3 49c c4 4c c4 4c 3c4 49c c3c4 e (64) The maximum order of covergece of optimal methods with five fuctios evaluatios is sixtee. To establish the order of covergece greater tha sixtee we have to display ext fiftee tedious terms, hece we have omitted them ad it is clear that the Kug ad Traub cojecture fails whe k. Fifth-order Polyomial equatio. k : Oe-poit third-order iterative method is give by x x r tr (6) ad the error equatio AEC c 3 c3 e (66). k : Oe-poit fourth-order iterative method is give by x x r tr t t r (67) ad the error equatio AEC c 3 4 cc3 c4 e (68) 3. k 3: Oe-poit fifth-order iterative method is give by Progressive Academic Publishig, UK Page 8

9 3 3 x x r tr t t r t tt t3 r (69) ad the error equatio AEC 3 4c 4 3c3 c c3 6cc4 c e (70) 4. k 4: Oe-poit sixth-order iterative method is give by x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 t4 r (7) ad the error equatio AEC 4 4c 3 6 8c c3 84c c3 7c3c4 8c c4 7cc e (7). k : Oe-poit seveth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r 4t t t t 6tt 3 t4 r 4t 8tt 84t t 7tt3 8t t3 7tt 4 r (73) ad the error equatio AEC 3c c3 80c c3 330c c3 4c4 0c c4 7cc3 c4 8c3c 36c c e (74) 6. k 6 : Oe-poit eighth-order iterative method is give by x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 t4 r t 8t t 84t t 7t t 8t t 7t t r t t 80t t 330t t 4t3 0t t3 7ttt3 8tt4 36t t4 r (7) ad the error equatio AEC 6 49c 6c c 990c c 87c c 49c c 4c c c 3c4 49c c3c4 9c4c 6c c 90cc 3c e (76) The maximum order of covergece of optimal methods with six fuctios evaluatios is thirty-two. To establish the order of covergece greater tha thirty-two we have to display ext thirty-oe tedious terms, hece we have omitted them ad it is clear that the Kug ad Traub cojecture fails whe k 3. Sixth-order Polyomial equatio. k : Oe-poit third-order iterative method is give by x x r tr (77) ad the error equatio AEC c 3 c3 e (78). k : Oe-poit fourth-order iterative method is give by Progressive Academic Publishig, UK Page 9

10 x x r tr t t r (79) ad the error equatio AEC c 3 4 cc3 c4 e (80) 3. k 3: Oe-poit fifth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r (8) ad the error equatio AEC 3 4c 4 3c3 c c3 6cc4 c e (8) 4. k 4: Oe-poit sixth-order iterative method is give by x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 t4 r (83) ad the error equatio AEC 4 4c 3 6 8c c3 84c c3 7c3c4 8c c4 7cc c6 e (84). k : Oe-poit seveth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r 4t t t t 6tt 3 t4 r 4t 8tt 84t t 7tt3 8t t3 7tt 4 t r (8) ad the error equatio AEC 3c c3 80c c3 330c c3 4c4 0c c4 7cc3 c4 8c3c 36c c 40cc 6 e (86) 6. k 6 : Oe-poit eighth-order iterative method is give by x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 t4 r t 8t t 84t t 7t t 8t t 7t t t r t t 80t t 330t t 4t3 0t t3 7ttt3 8tt4 36t t4 40tt r (87) ad the error equatio AEC 6 49c 6c c 990c c 87c c 49c c 4c c c 3c4 49c c3c4 9c4c 6c c 90cc 3c 4c 3c6 c c6 e (88) The maximum order of covergece of optimal methods with seve fuctios evaluatios is sixty-four. To establish the order of covergece greater tha sixty-four we have to display ext sixty-three tedious terms, hece we have omitted them ad it is clear that the Kug ad Traub cojecture fails whe k 63. Seveth-order Polyomial equatio. k : Oe-poit third-order iterative method is give by Progressive Academic Publishig, UK Page 0

11 x x r tr (89) ad the error equatio AEC c 3 c3 e (90). k : Oe-poit fourth-order iterative method is give by x x r tr t t r (9) ad the error equatio AEC c 3 4 cc3 c4 e (9) 3. k 3: Oe-poit fifth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r (93) ad the error equatio AEC 3 4c 4 3c3 c c3 6cc4 c e (94) 4. k 4: Oe-poit sixth-order iterative method is give by x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 t4 r (9) ad the error equatio AEC 4 4c 3 6 8c c3 84c c3 7c3c4 8c c4 7cc c6 e (96). k : Oe-poit seveth-order iterative method is give by 3 3 x x r tr t t r t tt t3 r 4t t t t 6tt 3 t4 r 4t 8tt 84t t 7tt3 8t t3 7tt 4 t r (97) ad the error equatio AEC 3c c3 80c c3 330c c3 4c4 0c c4 7cc3 c4 8c3c 36c c 40cc 6 e (98) 6. k 6 : Oe-poit eighth-order iterative method is give by x x r tr t t r t tt t3 r 4t 3t t t 6tt 3 t4 r t 8t t 84t t 7t t 8t t 7t t t r t t 80t t 330t t 4t3 0t t3 7ttt3 8tt4 36t t4 40tt r (99) ad the error equatio AEC 6 49c 6c c 990c c 87c c 49c c 4c c c 3c4 49c c3c4 9c4c 6c c 90cc 3c 4c 3c6 c c6 80c c7 e (00) Progressive Academic Publishig, UK Page

12 The maximum order of covergece of optimal methods with eight fuctios evaluatios is 8. To establish the order of covergece greater tha 8 we have to display ext 7 tedious term, hece we have omitted them ad it is clear that the Kug ad Traub cojecture fails whe k 7. The above procedure may be repeated for higher-order of polyomial equatio. Remark The ew oe-poit iterative method requires fuctio evaluatios ad has the order of covergece k. To determie the efficiecy idex of the ew method, defiitio shall be used. Hece, the efficiecy idex of the ew iterative method give by () is k ad the efficiecy idex of the classical Newto method is. For particular set of values of k, we have show that the efficiecy idex of the ew oe-poit method is much better tha the other similar methods which are based o the Kug ad Traub cojecture. Remark Kug ad Traub cojecture fails whe the total umber of terms required i the geeratig series (6) is give by d d (0) where d is the degree of the polyomial equatio. CONCLUSION I this work, a ew oe-poit k -order Newto-type method has bee preseted. The prime motive for presetig the ew class of iterative method was to exted ad demostrate the use of the Thukral s quadratic ad cubic equatio methods recetly itroduced i [8,9]. Furthermore, demostrate that the Kug ad Traub cojecture fails. It is very simple to evaluate the terms of the geeratig series (6), hece we ca costruct ay desire order of covergece. Recetly we have established the essetial advatages of the ew oe-poit method are: very high computatioal efficiecy; the ew method is ot limited to the Kug ad Traub cojecture; better efficiecy idex tha the classical Newto method; simple oestep iteratio, ot limited to quadratic ad cubic equatios [,,8,9], simple to costruct, efficiet ad robust. Fially, further ivestigatio is eeded to improve ad reduce the umber of terms required for the Kug ad Traub cojecture to fail. REFERENCES [] F. Ahmad, Higher order iterative methods for solvig matrix vector equatios (0), Researchgate, doi: 0.340/RG [] D. K. R. Babajee, O the Kug-Traub cojecture for iterative methods for solvig quadratic equatios, Algorithms 06, 9,, doi /a [3] W. Gautschi, Numerical Aalysis: a Itroductio, Birkhauser, 997. [4] H. Kug, J. F. Traub, Optimal order of two-poit ad multipoit iteratio, J. Assoc. Comput. Math. (974) [] M. S. Petkovic, B. Neta, L. D. Petkovic, J. Dzuic, Multipoit methods for solvig oliear equatios, Elsevier 0. Progressive Academic Publishig, UK Page

13 [6] A. M. Ostrowski, Solutios of equatios ad system of equatios, Academic Press, New York, 960. k -order covergece for solvig [7] R. Thukral, New improved Newto method with quadratic equatios, Euro. J. math. Computer. Sci. 3 () (06) 3-8. [8] R. Thukral, Aother Newto-type method with (k+) order covergece for solvig quadratic equatios, J. Adv. Math. Vol (9) (06) k -order covergece for [9] R. Thukral, New improved Newto-type method with solvig cubic equatios, Researchgate (06). [0] J. F. Traub, Iterative Methods for solutio of equatios, Chelsea publishig compay, New York 977. Progressive Academic Publishig, UK Page 3

Higher-order iterative methods by using Householder's method for solving certain nonlinear equations

Higher-order iterative methods by using Householder's method for solving certain nonlinear equations Math Sci Lett, No, 7- ( 7 Mathematical Sciece Letters A Iteratioal Joural http://dxdoiorg/785/msl/5 Higher-order iterative methods by usig Householder's method for solvig certai oliear equatios Waseem

More information

Ma 530 Introduction to Power Series

Ma 530 Introduction to Power Series Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power

More information

MATH 10550, EXAM 3 SOLUTIONS

MATH 10550, EXAM 3 SOLUTIONS MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,

More information

Some New Iterative Methods for Solving Nonlinear Equations

Some New Iterative Methods for Solving Nonlinear Equations World Applied Scieces Joural 0 (6): 870-874, 01 ISSN 1818-495 IDOSI Publicatios, 01 DOI: 10.589/idosi.wasj.01.0.06.830 Some New Iterative Methods for Solvig Noliear Equatios Muhammad Aslam Noor, Khalida

More information

Solving a Nonlinear Equation Using a New Two-Step Derivative Free Iterative Methods

Solving a Nonlinear Equation Using a New Two-Step Derivative Free Iterative Methods Applied ad Computatioal Mathematics 07; 6(6): 38-4 http://www.sciecepublishiggroup.com/j/acm doi: 0.648/j.acm.070606. ISSN: 38-5605 (Prit); ISSN: 38-563 (Olie) Solvig a Noliear Equatio Usig a New Two-Step

More information

Some Variants of Newton's Method with Fifth-Order and Fourth-Order Convergence for Solving Nonlinear Equations

Some Variants of Newton's Method with Fifth-Order and Fourth-Order Convergence for Solving Nonlinear Equations Copyright, Darbose Iteratioal Joural o Applied Mathematics ad Computatio Volume (), pp -6, 9 http//: ijamc.darbose.com Some Variats o Newto's Method with Fith-Order ad Fourth-Order Covergece or Solvig

More information

A New Family of Multipoint Iterative Methods for Finding Multiple Roots of Nonlinear Equations

A New Family of Multipoint Iterative Methods for Finding Multiple Roots of Nonlinear Equations Aerica Joural o Coputatioal ad Applied Matheatics 3, 3(3: 68-73 DOI:.593/j.ajca.333.3 A New Faily o Multipoit Iterative Methods or Fidig Multiple Roots o Noliear Equatios R. Thukral Padé Research Cetre,

More information

Z - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.

Z - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals. Z - Trasform The -trasform is a very importat tool i describig ad aalyig digital systems. It offers the techiques for digital filter desig ad frequecy aalysis of digital sigals. Defiitio of -trasform:

More information

MATH 31B: MIDTERM 2 REVIEW

MATH 31B: MIDTERM 2 REVIEW MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate x (x ) (x 3).. Partial Fractios Solutio: The umerator has degree less tha the deomiator, so we ca use partial fractios. Write x (x ) (x 3) = A x + A (x ) +

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

SOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.

SOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1. SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad

More information

CHAPTER 1 SEQUENCES AND INFINITE SERIES

CHAPTER 1 SEQUENCES AND INFINITE SERIES CHAPTER SEQUENCES AND INFINITE SERIES SEQUENCES AND INFINITE SERIES (0 meetigs) Sequeces ad limit of a sequece Mootoic ad bouded sequece Ifiite series of costat terms Ifiite series of positive terms Alteratig

More information

NUMERICAL METHODS FOR SOLVING EQUATIONS

NUMERICAL METHODS FOR SOLVING EQUATIONS Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:

More information

Numerical Solution of Non-linear Equations

Numerical Solution of Non-linear Equations Numerical Solutio of Noliear Equatios. INTRODUCTION The most commo reallife problems are oliear ad are ot ameable to be hadled by aalytical methods to obtai solutios of a variety of mathematical problems.

More information

The Phi Power Series

The Phi Power Series The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at http://semresearch.wordpress.com/.

More information

Newton Homotopy Solution for Nonlinear Equations Using Maple14. Abstract

Newton Homotopy Solution for Nonlinear Equations Using Maple14. Abstract Joural of Sciece ad Techology ISSN 9-860 Vol. No. December 0 Newto Homotopy Solutio for Noliear Equatios Usig Maple Nor Haim Abd. Rahma, Arsmah Ibrahim, Mohd Idris Jayes Faculty of Computer ad Mathematical

More information

Research Article A New Second-Order Iteration Method for Solving Nonlinear Equations

Research Article A New Second-Order Iteration Method for Solving Nonlinear Equations Abstract ad Applied Aalysis Volume 2013, Article ID 487062, 4 pages http://dx.doi.org/10.1155/2013/487062 Research Article A New Secod-Order Iteratio Method for Solvig Noliear Equatios Shi Mi Kag, 1 Arif

More information

New Twelfth Order Iterative Methods for Finding Multiple Roots of Nonlinear Equations

New Twelfth Order Iterative Methods for Finding Multiple Roots of Nonlinear Equations Aerica Joural o Coputatioal ad Applied Matheatics 05 5(: -4 DOI: 059/jajca050500 New Twelth Order Iterative Methods or Fidig Multiple Roots o Noliear Equatios R Thukral Padé Research Cetre 9 Deaswood Hill

More information

NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE

NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece

More information

IN many scientific and engineering applications, one often

IN many scientific and engineering applications, one often INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 5 Secod Degree Refiemet Jacobi Iteratio Method for Solvig System of Liear Equatio Tesfaye Kebede Abstract Several

More information

Applied Mathematics Letters

Applied Mathematics Letters Applied Mathematics Letters 5 (01) 03 030 Cotets lists available at SciVerse ScieceDirect Applied Mathematics Letters joural homepage: www.elsevier.com/locate/aml O ew computatioal local orders of covergece

More information

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0. THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of

More information

Recurrence Relations

Recurrence Relations Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The

More information

Chapter 10: Power Series

Chapter 10: Power Series Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because

More information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 Calculus II Basic Convergence Tests for Series MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

More information

The Riemann Zeta Function

The Riemann Zeta Function Physics 6A Witer 6 The Riema Zeta Fuctio I this ote, I will sketch some of the mai properties of the Riema zeta fuctio, ζ(x). For x >, we defie ζ(x) =, x >. () x = For x, this sum diverges. However, we

More information

Section 11.8: Power Series

Section 11.8: Power Series Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i

More information

f(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim

f(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =

More information

Math 113, Calculus II Winter 2007 Final Exam Solutions

Math 113, Calculus II Winter 2007 Final Exam Solutions Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this

More information

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =!

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =! .8,.9: Taylor ad Maclauri Series.8. Although we were able to fid power series represetatios for a limited group of fuctios i the previous sectio, it is ot immediately obvious whether ay give fuctio has

More information

ECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations

ECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows

More information

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled 1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

More information

NICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =

NICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) = AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,

More information

Two-step Extrapolated Newton s Method with High Efficiency Index

Two-step Extrapolated Newton s Method with High Efficiency Index Jour of Adv Research i Damical & Cotrol Systems Vol. 9 No. 017 Two-step Etrapolated Newto s Method with High Efficiecy Ide V.B. Kumar Vatti Dept. of Egieerig Mathematics Adhra Uiversity Visakhapatam Idia.

More information

Math 475, Problem Set #12: Answers

Math 475, Problem Set #12: Answers Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe

More information

Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or

Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad

More information

MATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and

MATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f

More information

The log-behavior of n p(n) and n p(n)/n

The log-behavior of n p(n) and n p(n)/n Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity

More information

Math 257: Finite difference methods

Math 257: Finite difference methods Math 257: Fiite differece methods 1 Fiite Differeces Remember the defiitio of a derivative f f(x + ) f(x) (x) = lim 0 Also recall Taylor s formula: (1) f(x + ) = f(x) + f (x) + 2 f (x) + 3 f (3) (x) +...

More information

Section A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics

Section A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics X0/70 NATIONAL QUALIFICATIONS 005 MONDAY, MAY.00 PM 4.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.

More information

Solution of Differential Equation from the Transform Technique

Solution of Differential Equation from the Transform Technique Iteratioal Joural of Computatioal Sciece ad Mathematics ISSN 0974-3189 Volume 3, Number 1 (2011), pp 121-125 Iteratioal Research Publicatio House http://wwwirphousecom Solutio of Differetial Equatio from

More information

Benaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco

Benaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco EXTENDING THE BERNOULLI-EULER METHOD FOR FINDING ZEROS OF HOLOMORPHIC FUNCTIONS Beaissa Beroussi Uiversité Abdelmalek Essaadi, ENSAT de Tager, B.P. 416, Tager, Morocco e-mail: Beaissa@fstt.ac.ma Mustapha

More information

Chapter 4. Fourier Series

Chapter 4. Fourier Series Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,

More information

Zeros of Polynomials

Zeros of Polynomials Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

More information

ENGI 9420 Lecture Notes 3 - Numerical Methods Page 3.01

ENGI 9420 Lecture Notes 3 - Numerical Methods Page 3.01 ENGI 940 Lecture Notes 3 - Numerical Methods Page 3.0 3. Numerical Methods The majority of equatios of iterest i actual practice do ot admit ay aalytic solutio. Eve equatios as simple as = e ad I = e d

More information

Notes on iteration and Newton s method. Iteration

Notes on iteration and Newton s method. Iteration Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f

More information

Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018

Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018 CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical

More information

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials

3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered

More information

Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations

Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said

More information

CHAPTER 10 INFINITE SEQUENCES AND SERIES

CHAPTER 10 INFINITE SEQUENCES AND SERIES CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece

More information

Solutions to Final Exam Review Problems

Solutions to Final Exam Review Problems . Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the

More information

MIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS

MIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will

More information

1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.

1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series. .3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(

More information

Math 113 Exam 4 Practice

Math 113 Exam 4 Practice Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for

More information

Properties and Tests of Zeros of Polynomial Functions

Properties and Tests of Zeros of Polynomial Functions Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by

More information

We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n

We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at

More information

f(w) w z =R z a 0 a n a nz n Liouville s theorem, we see that Q is constant, which implies that P is constant, which is a contradiction.

f(w) w z =R z a 0 a n a nz n Liouville s theorem, we see that Q is constant, which implies that P is constant, which is a contradiction. Theorem 3.6.4. [Liouville s Theorem] Every bouded etire fuctio is costat. Proof. Let f be a etire fuctio. Suppose that there is M R such that M for ay z C. The for ay z C ad R > 0 f (z) f(w) 2πi (w z)

More information

It is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.

It is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial. Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable

More information

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

More information

Recursive Algorithms. Recurrences. Recursive Algorithms Analysis

Recursive Algorithms. Recurrences. Recursive Algorithms Analysis Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects

More information

Physics 116A Solutions to Homework Set #1 Winter Boas, problem Use equation 1.8 to find a fraction describing

Physics 116A Solutions to Homework Set #1 Winter Boas, problem Use equation 1.8 to find a fraction describing Physics 6A Solutios to Homework Set # Witer 0. Boas, problem. 8 Use equatio.8 to fid a fractio describig 0.694444444... Start with the formula S = a, ad otice that we ca remove ay umber of r fiite decimals

More information

1 Generating functions for balls in boxes

1 Generating functions for balls in boxes Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

Taylor Series (BC Only)

Taylor Series (BC Only) Studet Study Sessio Taylor Series (BC Oly) Taylor series provide a way to fid a polyomial look-alike to a o-polyomial fuctio. This is doe by a specific formula show below (which should be memorized): Taylor

More information

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3 MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special

More information

Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number

Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios

More information

Taylor polynomial solution of difference equation with constant coefficients via time scales calculus

Taylor polynomial solution of difference equation with constant coefficients via time scales calculus TMSCI 3, o 3, 129-135 (2015) 129 ew Treds i Mathematical Scieces http://wwwtmscicom Taylor polyomial solutio of differece equatio with costat coefficiets via time scales calculus Veysel Fuat Hatipoglu

More information

MAT 271 Project: Partial Fractions for certain rational functions

MAT 271 Project: Partial Fractions for certain rational functions MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,

More information

5 Sequences and Series

5 Sequences and Series Bria E. Veitch 5 Sequeces ad Series 5. Sequeces A sequece is a list of umbers i a defiite order. a is the first term a 2 is the secod term a is the -th term The sequece {a, a 2, a 3,..., a,..., } is a

More information

CS321. Numerical Analysis and Computing

CS321. Numerical Analysis and Computing CS Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 September 8 5 What is the Root May physical system ca

More information

and each factor on the right is clearly greater than 1. which is a contradiction, so n must be prime.

and each factor on the right is clearly greater than 1. which is a contradiction, so n must be prime. MATH 324 Summer 200 Elemetary Number Theory Solutios to Assigmet 2 Due: Wedesday July 2, 200 Questio [p 74 #6] Show that o iteger of the form 3 + is a prime, other tha 2 = 3 + Solutio: If 3 + is a prime,

More information

INFINITE SEQUENCES AND SERIES

INFINITE SEQUENCES AND SERIES 11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS

More information

TR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT

TR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the

More information

Chapter 8. Euler s Gamma function

Chapter 8. Euler s Gamma function Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s that we will derive i the ext chapter. I the preset chapter we have collected some properties of the

More information

Sigma notation. 2.1 Introduction

Sigma notation. 2.1 Introduction Sigma otatio. Itroductio We use sigma otatio to idicate the summatio process whe we have several (or ifiitely may) terms to add up. You may have see sigma otatio i earlier courses. It is used to idicate

More information

A collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation

A collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation Iteratioal Joural of Mathematics Research. ISSN 0976-5840 Volume 9 Number 1 (017) pp. 45-51 Iteratioal Research Publicatio House http://www.irphouse.com A collocatio method for sigular itegral equatios

More information

Modified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations

Modified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations Noliear Aalysis ad Differetial Equatios, Vol. 5, 27, o. 4, 57-7 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/ade.27.62 Modified Decompositio Method by Adomia ad Rach for Solvig Noliear Volterra Itegro-

More information

On Divisibility concerning Binomial Coefficients

On Divisibility concerning Binomial Coefficients A talk give at the Natioal Chiao Tug Uiversity (Hsichu, Taiwa; August 5, 2010 O Divisibility cocerig Biomial Coefficiets Zhi-Wei Su Najig Uiversity Najig 210093, P. R. Chia zwsu@ju.edu.c http://math.ju.edu.c/

More information

Council for Innovative Research

Council for Innovative Research ABSTRACT ON ABEL CONVERGENT SERIES OF FUNCTIONS ERDAL GÜL AND MEHMET ALBAYRAK Yildiz Techical Uiversity, Departmet of Mathematics, 34210 Eseler, Istabul egul34@gmail.com mehmetalbayrak12@gmail.com I this

More information

Example 2. Find the upper bound for the remainder for the approximation from Example 1.

Example 2. Find the upper bound for the remainder for the approximation from Example 1. Lesso 8- Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute

More information

ENGI Series Page 6-01

ENGI Series Page 6-01 ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,

More information

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said

More information

PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) 5. (a) (b) (c) (d) (e) 7. (a) (b) (c) (d) (e)

PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) 5. (a) (b) (c) (d) (e) 7. (a) (b) (c) (d) (e) Math 0560, Exam 3 November 6, 07 The Hoor Code is i effect for this examiatio. All work is to be your ow. No calculators. The exam lasts for hour ad 5 mi. Be sure that your ame is o every page i case pages

More information

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f,

AP Calculus BC Review Applications of Derivatives (Chapter 4) and f, AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)

More information

Quiz. Use either the RATIO or ROOT TEST to determine whether the series is convergent or not.

Quiz. Use either the RATIO or ROOT TEST to determine whether the series is convergent or not. Quiz. Use either the RATIO or ROOT TEST to determie whether the series is coverget or ot. e .6 POWER SERIES Defiitio. A power series i about is a series of the form c 0 c a c a... c a... a 0 c a where

More information

MATH 1910 Workshop Solution

MATH 1910 Workshop Solution MATH 90 Workshop Solutio Fractals Itroductio: Fractals are atural pheomea or mathematical sets which exhibit (amog other properties) self similarity: o matter how much we zoom i, the structure remais the

More information

Math 210A Homework 1

Math 210A Homework 1 Math 0A Homework Edward Burkard Exercise. a) State the defiitio of a aalytic fuctio. b) What are the relatioships betwee aalytic fuctios ad the Cauchy-Riema equatios? Solutio. a) A fuctio f : G C is called

More information

Curve Sketching Handout #5 Topic Interpretation Rational Functions

Curve Sketching Handout #5 Topic Interpretation Rational Functions Curve Sketchig Hadout #5 Topic Iterpretatio Ratioal Fuctios A ratioal fuctio is a fuctio f that is a quotiet of two polyomials. I other words, p ( ) ( ) f is a ratioal fuctio if p ( ) ad q ( ) are polyomials

More information

arxiv: v3 [math.nt] 24 Dec 2017

arxiv: v3 [math.nt] 24 Dec 2017 DOUGALL S 5 F SUM AND THE WZ-ALGORITHM Abstract. We show how to prove the examples of a paper by Chu ad Zhag usig the WZ-algorithm. arxiv:6.085v [math.nt] Dec 07 Keywords. Geeralized hypergeometric series;

More information

INEQUALITIES BJORN POONEN

INEQUALITIES BJORN POONEN INEQUALITIES BJORN POONEN 1 The AM-GM iequality The most basic arithmetic mea-geometric mea (AM-GM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad

More information

Math 113 Exam 3 Practice

Math 113 Exam 3 Practice Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you

More information

Chapter 7: Numerical Series

Chapter 7: Numerical Series Chapter 7: Numerical Series Chapter 7 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals

More information

Sequences and Limits

Sequences and Limits Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q

More information

Fall 2013 MTH431/531 Real analysis Section Notes

Fall 2013 MTH431/531 Real analysis Section Notes Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters

More information

Shivley s Polynomials of Two Variables

Shivley s Polynomials of Two Variables It. Joural of Math. Aalysis, Vol. 6, 01, o. 36, 1757-176 Shivley s Polyomials of Two Variables R. K. Jaa, I. A. Salehbhai ad A. K. Shukla Departmet of Mathematics Sardar Vallabhbhai Natioal Istitute of

More information

AMS Mathematics Subject Classification : 40A05, 40A99, 42A10. Key words and phrases : Harmonic series, Fourier series. 1.

AMS Mathematics Subject Classification : 40A05, 40A99, 42A10. Key words and phrases : Harmonic series, Fourier series. 1. J. Appl. Math. & Computig Vol. x 00y), No. z, pp. A RECURSION FOR ALERNAING HARMONIC SERIES ÁRPÁD BÉNYI Abstract. We preset a coveiet recursive formula for the sums of alteratig harmoic series of odd order.

More information

Additional Notes on Power Series

Additional Notes on Power Series Additioal Notes o Power Series Mauela Girotti MATH 37-0 Advaced Calculus of oe variable Cotets Quick recall 2 Abel s Theorem 2 3 Differetiatio ad Itegratio of Power series 4 Quick recall We recall here

More information

Real Variables II Homework Set #5

Real Variables II Homework Set #5 Real Variables II Homework Set #5 Name: Due Friday /0 by 4pm (at GOS-4) Istructios: () Attach this page to the frot of your homework assigmet you tur i (or write each problem before your solutio). () Please

More information

Math 155 (Lecture 3)

Math 155 (Lecture 3) Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,

More information

Q-BINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA.

Q-BINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA. INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 Q-BINOMIALS AND THE GREATEST COMMON DIVISOR Keith R. Slavi 8474 SW Chevy Place, Beaverto, Orego 97008, USA slavi@dsl-oly.et Received:

More information