Explicit and closed formed solution of a differential equation. Closed form: since finite algebraic combination of. converges for x x0

Size: px
Start display at page:

Download "Explicit and closed formed solution of a differential equation. Closed form: since finite algebraic combination of. converges for x x0"

Transcription

1 Chapter 4 Series Solutios Epliit ad losed formed solutio of a differetial equatio y' y ; y() 3 ( ) ( 5 e ) y Closed form: sie fiite algebrai ombiatio of elemetary futios Series solutio: givig y ( ) as a ifiite series ivolvig ostat time powers of (Def 4.4) Power Series a overges for E6:!, whih overges for but diverges for ay

2 (Thm 4.5) overges for Suppose a series overges absolutely for all suh that.the the : Three possibilities for overgee of a. overge oly for. overge for all real umber 3. positive umber r, alled the radius of overgee of the series, suh that a diverges if r iterval of overgee. (Thm 4.6) overges if. r, r Suppose b for,,,... ad that r, is alled the ope lim b b L The b overges absolutely if L ad diverges if L. If L, the this test allows o olusio

3 Taylor ad Malauri Epasios i some iterval r, r f ( ) a f a f ' a f '' a (3) f 3 a3 ( k ) f k k a k k! ( ) ( k ) ak f This umber is alled the k th Taylor oeffiiet of f at! f The series is the Taylor series for f ( ) about If, the Taylor epasio is about ad is ofte alled a Malauri epasio e si!! are Malauri epasios 3

4 4. Power Series Solutios of Iitial Value Problems (Def 4.) Aalyti Futio: A futio f is aalyti at if f ( ) has a power series represetatio i some ope iterval about : f ( ) a ( ) i some iterval ( r, r). r is the radius of overgee ( 收斂半徑 ) (Thm 4.) Let p ad q be aalyti at. The the iitial value problem y ' py ( ) q ( ); y ( ) y has a solutio that is aalyti at That meas oeffiiets are aalyti at has a aalyti solutio at y ( ) a( ) ( ).! ( ) a y 4

5 E63: y ' e y ; y() 4 [ 解 ]: 5

6 (Thm 4.) Let p, q, ad f be aalyti at. The the iitial value problem y" py ( ) ' qy ( ) f( ) ; y ( ) A, y'( ) B has a uique solutio that is also aalyti at E64: y" y' e y 4; y(), y '() 4 [ 解 ]: 6

7 Eerise V: I eah problem, fid the first five ozero terms of the power series solutio of the iitial value problem, about the poit where the iitial oditios are give.. y" y' y ; y(), y '() { y } 3 6. y" y ; y() 3, y '() } { y 3 3. y" e y' y ; y() 3, y '() { y 34 } 6 3 7

8 4. Power Series Solutios Usig Reurree Relatios E65: y '' y,we wat a solutio epaded about [ 解 ]: y a y' a y'' a y ' begis at, y '' begis at Substitute yy, '' ito the differetial equatio '' y y a a a m m a m ( m, m ) m a am m ( m, m ) (m ) a a m a 3a a a 3 m 8

9 a a3 a a for =, 3,. a a a a a 43 4 a a a 54 5 for =, 3,. y a a a a a a a a a y, a y ' 9

10 E66: y '' y' 4 y [ 解 ]:

11 Eerise W: I eah problem, fid the reurree relatio ad use it to geerate the first five terms of the Malauri series of the geeral solutio.. y' y { y } a. y'' y' y { y ay ay, a y, a y' y , 5 3 y } 3. y y y '' ' ( ) { y a a a y, a y' }

12 4.3 Sigular Poits ad the Method of Forbeius '' ' P y Q y R y F ' y'' p y q y f (Def 4.) Ordiary ad Sigular Poits: is a ordiary poit of equatio '' ' if P y Q y R y F P ad Q/ P, R/ P,ad / is a sigular poit of equatio '' ' F P are aalyti at P y Q y R y F if is ot a ordiary poit E67: [ 解 ]: 3 y y y '' 5 ' 3

13 Oly fous o homogeeous equatio P y'' Q y' R y (Def 4.3) Regular ad Irregular Sigular Poits: is a regular sigular poit of equatio Q ad P P y'' Q y' R y if is a sigular poit, ad the futios R are aalyti at P. A sigular poit that is ot regular is said to be a irregular sigular poit E68: [ 解 ]: 3 y y y '' 5 ' 3 3

14 For regular sigular poit at y r is alled Forbeius series A Forbeius series eed ot be a power series, r may be egative or a oiteger Begis with ( ) r, whih is ostat oly if r y' r r This summatio for y ' begis at zero. Also true for y '' y'' r r r E69: y'' y' y [ 解 ]: 4

15 5

16 4.4 Seod Solutios (Thm 4.4) A seod solutio i the method of Frobeius: If ( r r ) is ot a iteger, two liearly idepedet Frobeius solutios: y ( ) r * r ad y ( ), * y ad y form a fudametal set o some iterval (, r) or (-r, ) 6

17 Eerise X: Fid all of the sigular poits ad lassify eah sigular poit as regular or irregular.. 3 y'' 4 6 y' y { Sigular poits are ad 3. is regular, 3 is regular} I eah problem, (a) show that zero is a regular sigular poit of the differetial equatio, (b) fid ad solve the idiial equatio, () determie the reurree relatio ad (d) use the results of (b) ad () to fid the first five ozero terms of two liearly idepedet Frobeius solutios.. 4 y'' y' y {(a) Q P ad R 4 are both aalyti at P so, is a regular sigular poit. (b) r r with roots r ad r () The reurree relatio is r r, 7

18 (d) Usig y r gives Usig r gives solutio 3. Q {(a) P 9 y'' y (b) y ad 9r 9r! }! R P ad oe solutio ad a seod, both aalyti at 9 with roots r 3 ad r 3 () 4, 3 3 r r (d) Usig r 3 gives ad 3 4 y 3! 47 3 Whereas r 3 gives ad 3 4 y 3! 58 3 } 8

After the completion of this section the student. V.4.2. Power Series Solution. V.4.3. The Method of Frobenius. V.4.4. Taylor Series Solution

After the completion of this section the student. V.4.2. Power Series Solution. V.4.3. The Method of Frobenius. V.4.4. Taylor Series Solution Chapter V ODE V.4 Power Series Solutio Otober, 8 385 V.4 Power Series Solutio Objetives: After the ompletio of this setio the studet - should reall the power series solutio of a liear ODE with variable

More information

Solutions 3.2-Page 215

Solutions 3.2-Page 215 Solutios.-Page Problem Fid the geeral solutios i powers of of the differetial equatios. State the reurree relatios ad the guarateed radius of overgee i eah ase. ) Substitutig,, ad ito the differetial equatio

More information

POWER SERIES METHODS CHAPTER 8 SECTION 8.1 INTRODUCTION AND REVIEW OF POWER SERIES

POWER SERIES METHODS CHAPTER 8 SECTION 8.1 INTRODUCTION AND REVIEW OF POWER SERIES CHAPTER 8 POWER SERIES METHODS SECTION 8. INTRODUCTION AND REVIEW OF POWER SERIES The power series method osists of substitutig a series y = ito a give differetial equatio i order to determie what the

More information

Bernoulli Numbers. n(n+1) = n(n+1)(2n+1) = n(n 1) 2

Bernoulli Numbers. n(n+1) = n(n+1)(2n+1) = n(n 1) 2 Beroulli Numbers Beroulli umbers are amed after the great Swiss mathematiia Jaob Beroulli5-705 who used these umbers i the power-sum problem. The power-sum problem is to fid a formula for the sum of the

More information

Calculus 2 TAYLOR SERIES CONVERGENCE AND TAYLOR REMAINDER

Calculus 2 TAYLOR SERIES CONVERGENCE AND TAYLOR REMAINDER Calulus TAYLO SEIES CONVEGENCE AND TAYLO EMAINDE Let the differee betwee f () ad its Taylor polyomial approimatio of order be (). f ( ) P ( ) + ( ) Cosider to be the remaider with the eat value ad the

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

Summation Method for Some Special Series Exactly

Summation Method for Some Special Series Exactly The Iteratioal Joural of Mathematis, Siee, Tehology ad Maagemet (ISSN : 39-85) Vol. Issue Summatio Method for Some Speial Series Eatly D.A.Gismalla Deptt. Of Mathematis & omputer Studies Faulty of Siee

More information

Calculus 2 - D. Yuen Final Exam Review (Version 11/22/2017. Please report any possible typos.)

Calculus 2 - D. Yuen Final Exam Review (Version 11/22/2017. Please report any possible typos.) Calculus - D Yue Fial Eam Review (Versio //7 Please report ay possible typos) NOTE: The review otes are oly o topics ot covered o previous eams See previous review sheets for summary of previous topics

More information

SOLUTION SET VI FOR FALL [(n + 2)(n + 1)a n+2 a n 1 ]x n = 0,

SOLUTION SET VI FOR FALL [(n + 2)(n + 1)a n+2 a n 1 ]x n = 0, 4. Series Solutios of Differetial Equatios:Special Fuctios 4.. Illustrative examples.. 5. Obtai the geeral solutio of each of the followig differetial equatios i terms of Maclauri series: d y (a dx = xy,

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

( 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

BESSEL EQUATION and BESSEL FUNCTIONS

BESSEL EQUATION and BESSEL FUNCTIONS BESSEL EQUATION ad BESSEL FUNCTIONS Bessel s Equatio Summary of Bessel Fuctios d y dy y d + d + =. If is a iteger, the two idepedet solutios of Bessel s Equatio are J J, Bessel fuctio of the first kid,

More information

9.3 Power Series: Taylor & Maclaurin Series

9.3 Power Series: Taylor & Maclaurin Series 9.3 Power Series: Taylor & Maclauri Series If is a variable, the a ifiite series of the form 0 is called a power series (cetered at 0 ). a a a a a 0 1 0 is a power series cetered at a c a a c a c a c 0

More information

Subject: Differential Equations & Mathematical Modeling -III. Lesson: Power series solutions of Differential Equations. about ordinary points

Subject: Differential Equations & Mathematical Modeling -III. Lesson: Power series solutions of Differential Equations. about ordinary points Power series solutio of Differetial equatios about ordiary poits Subject: Differetial Equatios & Mathematical Modelig -III Lesso: Power series solutios of Differetial Equatios about ordiary poits Lesso

More information

Power Series: A power series about the center, x = 0, is a function of x of the form

Power Series: A power series about the center, x = 0, is a function of x of the form You are familiar with polyomial fuctios, polyomial that has ifiitely may terms. 2 p ( ) a0 a a 2 a. A power series is just a Power Series: A power series about the ceter, = 0, is a fuctio of of the form

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

f x x c x c x c... x c...

f x x c x c x c... x c... CALCULUS BC WORKSHEET ON POWER SERIES. Derive the Taylor series formula by fillig i the blaks below. 4 5 Let f a a c a c a c a4 c a5 c a c What happes to this series if we let = c? f c so a Now differetiate

More information

Linear Differential Equations of Higher Order Basic Theory: Initial-Value Problems d y d y dy

Linear Differential Equations of Higher Order Basic Theory: Initial-Value Problems d y d y dy Liear Differetial Equatios of Higher Order Basic Theory: Iitial-Value Problems d y d y dy Solve: a( ) + a ( )... a ( ) a0( ) y g( ) + + + = d d d ( ) Subject to: y( 0) = y0, y ( 0) = y,..., y ( 0) = y

More information

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,

x x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0, Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative

More information

e to approximate (using 4

e to approximate (using 4 Review: Taylor Polyomials ad Power Series Fid the iterval of covergece for the series Fid a series for f ( ) d ad fid its iterval of covergece Let f( ) Let f arcta a) Fid the rd degree Maclauri polyomial

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

Sx [ ] = x must yield a

Sx [ ] = x must yield a Math -b Leture #5 Notes This wee we start with a remider about oordiates of a vetor relative to a basis for a subspae ad the importat speial ase where the subspae is all of R. This freedom to desribe vetors

More information

Chapter 2: Numerical Methods

Chapter 2: Numerical Methods Chapter : Numerical Methods. Some Numerical Methods for st Order ODEs I this sectio, a summar of essetial features of umerical methods related to solutios of ordiar differetial equatios is give. I geeral,

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

CHAPTER 5. Theory and Solution Using Matrix Techniques

CHAPTER 5. Theory and Solution Using Matrix Techniques A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL

More information

Subject: Differential Equations & Mathematical Modeling-III

Subject: Differential Equations & Mathematical Modeling-III Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso

More information

Honors Calculus Homework 13 Solutions, due 12/8/5

Honors Calculus Homework 13 Solutions, due 12/8/5 Hoors Calculus Homework Solutios, due /8/5 Questio Let a regio R i the plae be bouded by the curves y = 5 ad = 5y y. Sketch the regio R. The two curves meet where both equatios hold at oce, so where: y

More information

ENGI 9420 Engineering Analysis Assignment 3 Solutions

ENGI 9420 Engineering Analysis Assignment 3 Solutions ENGI 9 Egieerig Aalysis Assigmet Solutios Fall [Series solutio of ODEs, matri algebra; umerical methods; Chapters, ad ]. Fid a power series solutio about =, as far as the term i 7, to the ordiary differetial

More information

n 3 ln n n ln n is convergent by p-series for p = 2 > 1. n2 Therefore we can apply Limit Comparison Test to determine lutely convergent.

n 3 ln n n ln n is convergent by p-series for p = 2 > 1. n2 Therefore we can apply Limit Comparison Test to determine lutely convergent. 06 微甲 0-04 06-0 班期中考解答和評分標準. ( poits) Determie whether the series is absolutely coverget, coditioally coverget, or diverget. Please state the tests which you use. (a) ( poits) (b) ( poits) (c) ( poits)

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

f t dt. Write the third-degree Taylor polynomial for G

f t dt. Write the third-degree Taylor polynomial for G AP Calculus BC Homework - Chapter 8B Taylor, Maclauri, ad Power Series # Taylor & Maclauri Polyomials Critical Thikig Joural: (CTJ: 5 pts.) Discuss the followig questios i a paragraph: What does it mea

More information

M A T H F A L L CORRECTION. Algebra I 1 4 / 1 0 / U N I V E R S I T Y O F T O R O N T O

M A T H F A L L CORRECTION. Algebra I 1 4 / 1 0 / U N I V E R S I T Y O F T O R O N T O M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #4 CORRECTION Algebra I 1 4 / 1 0 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O P r o f e s s o r : D r o r B a r - N a t a Correctio Homework Assigmet

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

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

4x 2. (n+1) x 3 n+1. = lim. 4x 2 n+1 n3 n. n 4x 2 = lim = 3

4x 2. (n+1) x 3 n+1. = lim. 4x 2 n+1 n3 n. n 4x 2 = lim = 3 Exam Problems (x. Give the series (, fid the values of x for which this power series coverges. Also =0 state clearly what the radius of covergece is. We start by settig up the Ratio Test: x ( x x ( x x

More information

Ma 530 Infinite Series I

Ma 530 Infinite Series I Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li

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

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

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

Notes 19 Bessel Functions

Notes 19 Bessel Functions ECE 638 Fall 17 David R. Jackso Notes 19 Bessel Fuctios Notes are from D. R. Wilto, Dept. of ECE 1 Cylidrical Wave Fuctios Helmholtz equatio: ψ + k ψ = I cylidrical coordiates: ψ 1 ψ 1 ψ ψ ρ ρ ρ ρ φ z

More information

IIT JAM Mathematical Statistics (MS) 2006 SECTION A

IIT JAM Mathematical Statistics (MS) 2006 SECTION A IIT JAM Mathematical Statistics (MS) 6 SECTION A. If a > for ad lim a / L >, the which of the followig series is ot coverget? (a) (b) (c) (d) (d) = = a = a = a a + / a lim a a / + = lim a / a / + = lim

More information

Ans: a n = 3 + ( 1) n Determine whether the sequence converges or diverges. If it converges, find the limit.

Ans: a n = 3 + ( 1) n Determine whether the sequence converges or diverges. If it converges, find the limit. . Fid a formula for the term a of the give sequece: {, 3, 9, 7, 8 },... As: a = 3 b. { 4, 9, 36, 45 },... As: a = ( ) ( + ) c. {5,, 5,, 5,, 5,,... } As: a = 3 + ( ) +. Determie whether the sequece coverges

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

8. Applications To Linear Differential Equations

8. Applications To Linear Differential Equations 8. Applicatios To Liear Differetial Equatios 8.. Itroductio 8.. Review Of Results Cocerig Liear Differetial Equatios Of First Ad Secod Orders 8.3. Eercises 8.4. Liear Differetial Equatios Of Order N 8.5.

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

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

A brief introduction to linear algebra

A brief introduction to linear algebra CHAPTER 6 A brief itroductio to liear algebra 1. Vector spaces ad liear maps I what follows, fix K 2{Q, R, C}. More geerally, K ca be ay field. 1.1. Vector spaces. Motivated by our ituitio of addig ad

More information

(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)

(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b) Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig

More information

Linearly Independent Sets, Bases. Review. Remarks. A set of vectors,,, in a vector space is said to be linearly independent if the vector equation

Linearly Independent Sets, Bases. Review. Remarks. A set of vectors,,, in a vector space is said to be linearly independent if the vector equation Liearly Idepedet Sets Bases p p c c p Review { v v vp} A set of vectors i a vector space is said to be liearly idepedet if the vector equatio cv + c v + + c has oly the trivial solutio = = { v v vp} The

More information

Certain inclusion properties of subclass of starlike and convex functions of positive order involving Hohlov operator

Certain inclusion properties of subclass of starlike and convex functions of positive order involving Hohlov operator Iteratioal Joural of Pure ad Applied Mathematial Siees. ISSN 0972-9828 Volume 0, Number (207), pp. 85-97 Researh Idia Publiatios http://www.ripubliatio.om Certai ilusio properties of sublass of starlike

More information

Random Models. Tusheng Zhang. February 14, 2013

Random Models. Tusheng Zhang. February 14, 2013 Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the

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

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

Topic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist.

Topic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist. Topic 5 [44 marks] 1a (i) Fid the rage of values of for which eists 1 Write dow the value of i terms of 1, whe it does eist Fid the solutio to the differetial equatio 1b give that y = 1 whe = π (cos si

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

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

Topic 9 - Taylor and MacLaurin Series

Topic 9 - Taylor and MacLaurin Series Topic 9 - Taylor ad MacLauri Series A. Taylors Theorem. The use o power series is very commo i uctioal aalysis i act may useul ad commoly used uctios ca be writte as a power series ad this remarkable result

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

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) is empty. B) is a finite set. C) can be a countably infinite set. D) can be an uncountable set.

A) is empty. B) is a finite set. C) can be a countably infinite set. D) can be an uncountable set. M.A./M.Sc. (Mathematics) Etrace Examiatio 016-17 Max Time: hours Max Marks: 150 Istructios: There are 50 questios. Every questio has four choices of which exactly oe is correct. For correct aswer, 3 marks

More information

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges

More information

Chapter 1. Complex Numbers. Dr. Pulak Sahoo

Chapter 1. Complex Numbers. Dr. Pulak Sahoo Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler

More information

SOME NOTES ON INEQUALITIES

SOME NOTES ON INEQUALITIES SOME NOTES ON INEQUALITIES Rihard Hoshio Here are four theorems that might really be useful whe you re workig o a Olympiad problem that ivolves iequalities There are a buh of obsure oes Chebyheff, Holder,

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

Notes #3 Sequences Limit Theorems Monotone and Subsequences Bolzano-WeierstraßTheorem Limsup & Liminf of Sequences Cauchy Sequences and Completeness

Notes #3 Sequences Limit Theorems Monotone and Subsequences Bolzano-WeierstraßTheorem Limsup & Liminf of Sequences Cauchy Sequences and Completeness Notes #3 Sequeces Limit Theorems Mootoe ad Subsequeces Bolzao-WeierstraßTheorem Limsup & Limif of Sequeces Cauchy Sequeces ad Completeess This sectio of otes focuses o some of the basics of sequeces of

More information

Series: Infinite Sums

Series: Infinite Sums Series: Ifiite Sums Series are a way to mae sese of certai types of ifiitely log sums. We will eed to be able to do this if we are to attai our goal of approximatig trascedetal fuctios by usig ifiite degree

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

AH Checklist (Unit 3) AH Checklist (Unit 3) Matrices

AH Checklist (Unit 3) AH Checklist (Unit 3) Matrices AH Checklist (Uit 3) AH Checklist (Uit 3) Matrices Skill Achieved? Kow that a matrix is a rectagular array of umbers (aka etries or elemets) i paretheses, each etry beig i a particular row ad colum Kow

More information

Practice Problems: Taylor and Maclaurin Series

Practice Problems: Taylor and Maclaurin Series Practice Problems: Taylor ad Maclauri Series Aswers. a) Start by takig derivatives util a patter develops that lets you to write a geeral formula for the -th derivative. Do t simplify as you go, because

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

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

CONTENTS. Course Goals. Course Materials Lecture Notes:

CONTENTS. Course Goals. Course Materials Lecture Notes: INTRODUCTION Ho Chi Mih City OF Uiversity ENVIRONMENTAL of Techology DESIGN Faculty Chapter of Civil 1: Orietatio. Egieerig Evaluatio Departmet of mathematical of Water Resources skill Egieerig & Maagemet

More information

+ {JEE Advace 03} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks: 00. If A (α, β) = (a) A( α, β) = A( α, β) (c) Adj (A ( α, β)) = Sol : We

More information

University of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck!

University of Colorado Denver Dept. Math. & Stat. Sciences Applied Analysis Preliminary Exam 13 January 2012, 10:00 am 2:00 pm. Good luck! Uiversity of Colorado Dever Dept. Math. & Stat. Scieces Applied Aalysis Prelimiary Exam 13 Jauary 01, 10:00 am :00 pm Name: The proctor will let you read the followig coditios before the exam begis, ad

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

Digital Signal Processing. Homework 2 Solution. Due Monday 4 October Following the method on page 38, the difference equation

Digital Signal Processing. Homework 2 Solution. Due Monday 4 October Following the method on page 38, the difference equation Digital Sigal Proessig Homework Solutio Due Moda 4 Otober 00. Problem.4 Followig the method o page, the differee equatio [] (/4[-] + (/[-] x[-] has oeffiiets a0, a -/4, a /, ad b. For these oeffiiets A(z

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

Numerical Methods for Ordinary Differential Equations

Numerical Methods for Ordinary Differential Equations Numerical Methods for Ordiary Differetial Equatios Braislav K. Nikolić Departmet of Physics ad Astroomy, Uiversity of Delaware, U.S.A. PHYS 460/660: Computatioal Methods of Physics http://www.physics.udel.edu/~bikolic/teachig/phys660/phys660.html

More information

MATHEMATICS Code No. 13 INSTRUCTIONS

MATHEMATICS Code No. 13 INSTRUCTIONS DO NOT OPEN THIS TEST BOOKLET UNTIL YOU ARE ASKED TO DO SO COMBINED COMPETITIVE (PRELIMINARY) EXAMINATION, 0 Serial No. MATHEMATICS Code No. A Time Allowed : Two Hours Maimum Marks : 00 INSTRUCTIONS. IMMEDIATELY

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

Chapter 10 Partial Differential Equations and Fourier Series

Chapter 10 Partial Differential Equations and Fourier Series Math-33 Chapter Partial Differetial Equatios November 6, 7 Chapter Partial Differetial Equatios ad Fourier Series Math-33 Chapter Partial Differetial Equatios November 6, 7. Boudary Value Problems for

More information

Sequences, Sums, and Products

Sequences, Sums, and Products CSCE 222 Discrete Structures for Computig Sequeces, Sums, ad Products Dr. Philip C. Ritchey Sequeces A sequece is a fuctio from a subset of the itegers to a set S. A discrete structure used to represet

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

TAYLOR AND MACLAURIN SERIES

TAYLOR AND MACLAURIN SERIES Calculus TAYLOR AND MACLAURIN SERIES Give a uctio ( ad a poit a, we wish to approimate ( i the eighborhood o a by a polyomial o degree. c c ( a c( a c( a P ( c ( a We have coeiciets to choose. We require

More information

CALCULUS IN A NUTSHELL INTRODUCTION:

CALCULUS IN A NUTSHELL INTRODUCTION: CALCULUS IN A NUTSHELL INTRODUCTION: Studets are usually itroduced to basic calculus i twelfth grade i high school or the first year of college. The course is typically stretched out over oe year ad ivolves

More information

, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx)

, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx) Cosider the differetial equatio y '' k y 0 has particular solutios y1 si( kx) ad y cos( kx) I geeral, ay liear combiatio of y1 ad y, cy 1 1 cy where c1, c is also a solutio to the equatio above The reaso

More information

In number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.

In number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play. Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers

More information

Chimica Inorganica 3

Chimica Inorganica 3 himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule

More information

Partial Differential Equations

Partial Differential Equations EE 84 Matematical Metods i Egieerig Partial Differetial Eqatios Followig are some classical partial differetial eqatios were is assmed to be a fctio of two or more variables t (time) ad y (spatial coordiates).

More information

Math 21C Brian Osserman Practice Exam 2

Math 21C Brian Osserman Practice Exam 2 Math 1C Bria Osserma Practice Exam 1 (15 pts.) Determie the radius ad iterval of covergece of the power series (x ) +1. First we use the root test to determie for which values of x the series coverges

More information

Mathematical Description of Discrete-Time Signals. 9/10/16 M. J. Roberts - All Rights Reserved 1

Mathematical Description of Discrete-Time Signals. 9/10/16 M. J. Roberts - All Rights Reserved 1 Mathematical Descriptio of Discrete-Time Sigals 9/10/16 M. J. Roberts - All Rights Reserved 1 Samplig ad Discrete Time Samplig is the acquisitio of the values of a cotiuous-time sigal at discrete poits

More information

PAPER : IIT-JAM 2010

PAPER : IIT-JAM 2010 MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure

More information

Math 128A: Homework 1 Solutions

Math 128A: Homework 1 Solutions Math 8A: Homework Solutios Due: Jue. Determie the limits of the followig sequeces as. a) a = +. lim a + = lim =. b) a = + ). c) a = si4 +6) +. lim a = lim = lim + ) [ + ) ] = [ e ] = e 6. Observe that

More information

Mathematical Series (You Should Know)

Mathematical Series (You Should Know) Mathematical Series You Should Kow Mathematical series represetatios are very useful tools for describig images or for solvig/approimatig the solutios to imagig problems. The may be used to epad a fuctio

More information

L 5 & 6: RelHydro/Basel. f(x)= ( ) f( ) ( ) ( ) ( ) n! 1! 2! 3! If the TE of f(x)= sin(x) around x 0 is: sin(x) = x - 3! 5!

L 5 & 6: RelHydro/Basel. f(x)= ( ) f( ) ( ) ( ) ( ) n! 1! 2! 3! If the TE of f(x)= sin(x) around x 0 is: sin(x) = x - 3! 5! aylor epasio: Let ƒ() be a ifiitely differetiable real fuctio. At ay poit i the eighbourhood of =0, the fuctio ca be represeted as a power series of the followig form: X 0 f(a) f() ƒ() f()= ( ) f( ) (

More information

SCORE. Exam 2. MA 114 Exam 2 Fall 2017

SCORE. Exam 2. MA 114 Exam 2 Fall 2017 Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use a graphig calculator

More information

I PUC MATHEMATICS CHAPTER - 08 Binomial Theorem. x 1. Expand x + using binomial theorem and hence find the coefficient of

I PUC MATHEMATICS CHAPTER - 08 Binomial Theorem. x 1. Expand x + using binomial theorem and hence find the coefficient of Two Maks Questios I PU MATHEMATIS HAPTER - 08 Biomial Theoem. Epad + usig biomial theoem ad hece fid the coefficiet of y y. Epad usig biomial theoem. Hece fid the costat tem of the epasio.. Simplify +

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

Inverse Matrix. A meaning that matrix B is an inverse of matrix A.

Inverse Matrix. A meaning that matrix B is an inverse of matrix A. Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix

More information