# 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 +

Size: px
Start display at page:

Download "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 +"

Transcription

1 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 the series I geeral, the series will coverge or diverge, depedig o the choice of x. The power series always coverges for x = 0 to the umber c 0. Example 98. For what values of x does the power series series x / coverge? Solutio: By the root test, x = x x as

2 62. POWER SERIES 125 So, the series coverges for all 1 < x < 1 ad diverges as x > 1 or x < 1. The root test is icoclusive for x = ±1. These values have to ivestigated by differet meas. For x = 1, the power series becomes the harmoic series 1/ which is diverget. For x = 1, the power series becomes the alteratig harmoic series ( 1) / which is coverget. Thus, the power series coverges if x [ 1, 1) ad diverges otherwise. Give a umber a, cosider a power series i the variable y = x a c y = c (x a) It is also called a power series cetered at a or a power series about a. Let S be the set of all values of x for which a power series i x coverges ad let S a be the set of all values of x for which the correspodig power series i (x a) coverges. What is the relatio betwee S ad S a? Sice the series are obtaied from oe aother by merely shiftig the value of the variable by a umber a, x x a, the set S a is therefore obtaied by addig the umber a to every elemet of S: (60) x S a x a S = S a = {x x a S} For example, the series (x 2) / coverges if x 2 [ 1, 1) or x [1, 3) ad diverges otherwise by Example 98. Thus, the problem of fidig the set S a is equivalet to the problem of fidig the set S Power series as a fuctio. Suppose that a power series i x coverges o a set S. The it defies a fuctio o S: f(x) = c x, x S The set S is called the domai of such a fuctio. Fuctios defied by power series are most commo i applicatios. May of them have special otatios (like elemetary fuctios si, cos, exp, etc). Their properties are well studied. I what follows it will be show that familiar elemetary fuctios si(x), cos(x), exp(x), etc ca also be represeted as power series. Example 99. Fid the domai of the Bessel fuctio of order 0 that is defied by the power series J 0 (x) = ( 1) 2 2 (!) 2 x2

3 SEQUENCES AND SERIES where, by commo covetio, 0! = 1. Solutio: Sice a = c x 2 cotais the factorial, the ratio test is more coveiet: a +1 a = x 2 c +1 c = x (!) 2 2 2(+1) (( + 1)!) 2 = x ( + 1) 2 0 as. So, the series coverges for all x. Values of a fuctio defied by a power series ca be estimated by partial sums which are polyomials i the variable x: f(x) f (x) = c k x k = c 0 + c 1 x + c 2 x c x k=0 Thus, partial sums defie a sequece of polyomials that coverges to the fuctio o S, f (x) f(x) for all x S. The accuracy of the approximatio is determied by the remaider R (x) = f(x) f (x). The accuracy assessmet is discussed i Sectio Sice the remider R (x) is a fuctio o S, the error of the approximatio is ot geerally uiform, i.e. it depeds o x Radius of covergece. The set S o which a power series is coverget is a importat characteristic ad its properties have to be studied. Lemma 2. (Properties of a power series) (i). If a power series c x coverges whe x = b 0, the it coverges wheever x b. (ii). If a power series c x diverges whe x = d 0, the it diverges wheever x d. Proof: If c b coverges, the by the ecessary coditio for covergece, c b 0 as. This meas, i particular, that for ε = 1 there exists a iteger N such that c b < ε = 1 for all > N. Thus, for > N, c x = c b x = c b x < x b b b which shows that the series c x coverges by compariso with the geometric series q where q = x/b ad x/b < 1 or x < b. Suppose that c d diverges. If x is ay umber such that x > d, the c x caot coverge because, by part (i) of the lemma, the covergece of c x implies the covergece of c d. Therefore c x diverges. This lemma allows us to establish the followig descriptio of the set S.

4 62. POWER SERIES 127 Theorem 45. (Covergece properties of a power series) For a power series c x there are oly three possibilities: 1. The series coverges oly whe x = 0 2. The series coverges for all x 3. There is a positive umber R such that the series coverges if x < R ad diverges if x > R. Proof: Suppose that either case 1 or case 2 is true. There there are umbers b 0 ad d 0 such that the power series coverges for x = b ad diverges for x = d. By Lemma 2 the set of covergece S lies i the iterval: x d for all x S. This shows that d is a upper boud for the set S. By the completeess axiom, S has a least upper boud R = sup S. If x > R, the x S ad c x diverges. If x < R, the x is ot a upper boud for S ad there exist a umber b S such that b > x. Sice b S, c x coverges by Lemma 2. Theorem 45 shows that a power series coverges i a sigle ope iterval ( R, R) ad diverges outside this iterval. The set S may or may ot iclude the poits x = ±R. This questio requires a special ivestigatio just like i Example 98. So, the umber R is characteristic for covergece properties of a power series. Defiitio 17. (Radius of covergece) The radius of covergece of a power series c x is a positive umber R > 0 such that the series coverges i the ope iterval ( R, R) ad diverges outside it. A power series is said to have a zero radius of covergece, R = 0, if it coverges oly whe x = 0. A power series is said to have a ifiite radius of covergece, R =, if it coverges for all values of x. The ratio or root test ca be used to determie the radius of covergece. Corollary 3. (Radius of covergece of a power series) Give a power series c x, if if c +1 lim c lim = α = R = 1 α c = α = R = 1 α where R = 0 if α = ad R = if α = 0 Proof: Put a = c x i the ratio test (Theorem 60.4). The a +1 / a = x c +1 / c x α. The series coverges if x α < 1 which shows that R = 1/α. Similarly, usig the root test (Theorem 40), a = x c x α < 1, which shows that R = 1/α.

5 SEQUENCES AND SERIES Remark. If the sequeces i Corollary 3 do ot coverge, the Theorem 41 should be used where a = c x. Oce the radius of covergece has bee foud ad 0 < R <, the cases x = ±R have to be ivestigated by some other meas (as the root or ratio test is icoclusive i this case) to determie the iterval of covergece S of a power series. Example 100. Fid the radius of covergece ad the iterval of covergece of the power series c x where c = ( q) / + 1 ad q > 0 Solutio: c +1 c = q q = q = q 1 + 1/ 1 + 2/ q = α Therefore R = 1/α = 1/q. If x = 1/q, the c x = ( 1) / + 1 = ( 1) b. The sequece b coverges mootoically to zero so that ( 1) b coverges by the alteratig series test. If x = 1/q, the c x = 1/ + 1 > 1/ 2, 1. The p series 1/ 1/2 diverges (p = 1/2 < 1) so that 1/ + 1 diverges by the compariso test. Thus, the iterval of covergece is S = [ 1/q, 1/q). Example 101. Fid the radius of covergece ad the iterval of covergece of the power series 2 (x + 1) /q where q > 0. Solutio: Put y = x +1. If S is the iterval of covergece of c y where c = 2 /q the the iterval of covergece i questio is obtaied by addig 1 to all umbers i S accordig to the rule (60). c = 1 q 2 = 1 q ( ) 2 1 q = α So R = 1/α = q. If y = q, c y = 2 ad the series 2 diverges (a = 2 does ot coverge to 0). If y = q, the c x = ( 1) 2 ad the series diverges because a = ( 1) 2 does ot coverge to 0. The series coverges oly if y = x + 1 < q ad, hece, the iterval of covergece is x ( q 1, q 1) (the iterval ( q, q) shifted by 1).

### sin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =

60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece

### Chapter 6 Infinite Series

Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat

### 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

### SUMMARY OF SEQUENCES AND SERIES

SUMMARY OF SEQUENCES AND SERIES Importat Defiitios, Results ad Theorems for Sequeces ad Series Defiitio. A sequece {a } has a limit L ad we write lim a = L if for every ɛ > 0, there is a correspodig iteger

### Arkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan

Arkasas Tech Uiversity MATH 94: Calculus II Dr Marcel B Fia 85 Power Series Let {a } =0 be a sequece of umbers The a power series about x = a is a series of the form a (x a) = a 0 + a (x a) + a (x a) +

### 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

### n=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n

Series. Defiitios ad first properties A series is a ifiite sum a + a + a +..., deoted i short by a. The sequece of partial sums of the series a is the sequece s ) defied by s = a k = a +... + a,. k= Defiitio

### 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

### 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

### Section 5.5. Infinite Series: The Ratio Test

Differece Equatios to Differetial Equatios Sectio 5.5 Ifiite Series: The Ratio Test I the last sectio we saw that we could demostrate the covergece of a series a, where a 0 for all, by showig that a approaches

### 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

### 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

### 1 Lecture 2: Sequence, Series and power series (8/14/2012)

Summer Jump-Start Program for Aalysis, 202 Sog-Yig Li Lecture 2: Sequece, Series ad power series (8/4/202). More o sequeces Example.. Let {x } ad {y } be two bouded sequeces. Show lim sup (x + y ) lim

### 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

### SCORE. Exam 2. MA 114 Exam 2 Fall 2016

MA 4 Exam Fall 06 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

### MAS111 Convergence and Continuity

MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece

### Convergence: nth-term Test, Comparing Non-negative Series, Ratio Test

Covergece: th-term Test, Comparig No-egative Series, Ratio Test Power Series ad Covergece We have writte statemets like: l + x = x x2 + x3 2 3 + x + But we have ot talked i depth about what values of x

### MTH 122 Calculus II Essex County College Division of Mathematics and Physics 1 Lecture Notes #20 Sakai Web Project Material

MTH 1 Calculus II Essex Couty College Divisio of Mathematics ad Physics 1 Lecture Notes #0 Sakai Web Project Material 1 Power Series 1 A power series is a series of the form a x = a 0 + a 1 x + a x + a

### 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

### M17 MAT25-21 HOMEWORK 5 SOLUTIONS

M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series

### 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

### SCORE. Exam 2. MA 114 Exam 2 Fall 2016

Exam 2 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

### INFINITE SEQUENCES AND SERIES

INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES I geeral, it is difficult to fid the exact sum of a series. We were able to accomplish this for geometric series ad the series /[(+)]. This is

### 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

### Sequences. Notation. Convergence of a Sequence

Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it

### Math 113 Exam 3 Practice

Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This

### 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

### ( 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

### 7 Sequences of real numbers

40 7 Sequeces of real umbers 7. Defiitios ad examples Defiitio 7... A sequece of real umbers is a real fuctio whose domai is the set N of atural umbers. Let s : N R be a sequece. The the values of s are

### 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

### 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

### 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

### 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

### 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

### 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

### Are the following series absolutely convergent? n=1. n 3. n=1 n. ( 1) n. n=1 n=1

Absolute covergece Defiitio A series P a is called absolutely coverget if the series of absolute values P a is coverget. If the terms of the series a are positive, absolute covergece is the same as covergece.

### Sec 8.4. Alternating Series Test. A. Before Class Video Examples. Example 1: Determine whether the following series is convergent or divergent.

Sec 8.4 Alteratig Series Test A. Before Class Video Examples Example 1: Determie whether the followig series is coverget or diverget. a) ( 1)+1 =1 b) ( 1) 2 1 =1 Example 2: Determie whether the followig

### BC: Q401.CH9A Convergent and Divergent Series (LESSON 1)

BC: Q40.CH9A Coverget ad Diverget Series (LESSON ) INTRODUCTION Sequece Notatio: a, a 3, a,, a, Defiitio: A sequece is a fuctio f whose domai is the set of positive itegers. Defiitio: A ifiite series (or

### Math 341 Lecture #31 6.5: Power Series

Math 341 Lecture #31 6.5: Power Series We ow tur our attetio to a particular kid of series of fuctios, amely, power series, f(x = a x = a 0 + a 1 x + a 2 x 2 + where a R for all N. I terms of a series

### 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

### 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

### AP Calculus Chapter 9: Infinite Series

AP Calculus Chapter 9: Ifiite Series 9. Sequeces a, a 2, a 3, a 4, a 5,... Sequece: A fuctio whose domai is the set of positive itegers = 2 3 4 a = a a 2 a 3 a 4 terms of the sequece Begi with the patter

### Alternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n.

0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Sectio 9.5 Alteratig Series Use the Alteratig Series Test to determie whether a ifiite series coverges. Use the Alteratig Series Remaider to approximate

### 5.6 Absolute Convergence and The Ratio and Root Tests

5.6 Absolute Covergece ad The Ratio ad Root Tests Bria E. Veitch 5.6 Absolute Covergece ad The Ratio ad Root Tests Recall from our previous sectio that diverged but ( ) coverged. Both of these sequeces

### 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

### 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

### Series Review. a i converges if lim. i=1. a i. lim S n = lim i=1. 2 k(k + 2) converges. k=1. k=1

Defiitio: We say that the series S = Series Review i= a i is the sum of the first terms. i= a i coverges if lim S exists ad is fiite, where The above is the defiitio of covergece for series. order to see

### 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

### (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

### 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

### d) If the sequence of partial sums converges to a limit L, we say that the series converges and its

Ifiite Series. Defiitios & covergece Defiitio... Let {a } be a sequece of real umbers. a) A expressio of the form a + a +... + a +... is called a ifiite series. b) The umber a is called as the th term

### 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

### Mathematical Methods for Physics and Engineering

Mathematical Methods for Physics ad Egieerig Lecture otes Sergei V. Shabaov Departmet of Mathematics, Uiversity of Florida, Gaiesville, FL 326 USA CHAPTER The theory of covergece. Numerical sequeces..

### Mathematics 116 HWK 21 Solutions 8.2 p580

Mathematics 6 HWK Solutios 8. p580 A abbreviatio: iff is a abbreviatio for if ad oly if. Geometric Series: Several of these problems use what we worked out i class cocerig the geometric series, which I

### Math 112 Fall 2018 Lab 8

Ma Fall 08 Lab 8 Sums of Coverget Series I Itroductio Ofte e fuctios used i e scieces are defied as ifiite series Determiig e covergece or divergece of a series becomes importat ad it is helpful if e sum

### 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,

### Chapter 6 Overview: Sequences and Numerical Series. For the purposes of AP, this topic is broken into four basic subtopics:

Chapter 6 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 (which is what most studets

### 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

### 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

### MA131 - Analysis 1. Workbook 9 Series III

MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................

### LECTURE SERIES WITH NONNEGATIVE TERMS (II). SERIES WITH ARBITRARY TERMS

LECTURE 4 SERIES WITH NONNEGATIVE TERMS II). SERIES WITH ARBITRARY TERMS Series with oegative terms II) Theorem 4.1 Kummer s Test) Let x be a series with positive terms. 1 If c ) N i 0, + ), r > 0 ad 0

### Sequence A sequence is a function whose domain of definition is the set of natural numbers.

Chapter Sequeces Course Title: Real Aalysis Course Code: MTH3 Course istructor: Dr Atiq ur Rehma Class: MSc-I Course URL: wwwmathcityorg/atiq/fa8-mth3 Sequeces form a importat compoet of Mathematical Aalysis

### The Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1

460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that

### Sequences. A Sequence is a list of numbers written in order.

Sequeces A Sequece is a list of umbers writte i order. {a, a 2, a 3,... } The sequece may be ifiite. The th term of the sequece is the th umber o the list. O the list above a = st term, a 2 = 2 d term,

### Read carefully the instructions on the answer book and make sure that the particulars required are entered on each answer book.

THE UNIVERSITY OF WARWICK FIRST YEAR EXAMINATION: Jauary 2009 Aalysis I Time Allowed:.5 hours Read carefully the istructios o the aswer book ad make sure that the particulars required are etered o each

### 10.6 ALTERNATING SERIES

0.6 Alteratig Series Cotemporary Calculus 0.6 ALTERNATING SERIES I the last two sectios we cosidered tests for the covergece of series whose terms were all positive. I this sectio we examie series whose

### MATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.

MATH 1080: Calculus of Oe Variable II Fall 2017 Textbook: Sigle Variable Calculus: Early Trascedetals, 7e, by James Stewart Uit 3 Skill Set Importat: Studets should expect test questios that require a

### 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

### Chapter 6: Numerical Series

Chapter 6: Numerical Series 327 Chapter 6 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

### 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

### Carleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below.

Carleto College, Witer 207 Math 2, Practice Fial Prof. Joes Note: the exam will have a sectio of true-false questios, like the oe below.. True or False. Briefly explai your aswer. A icorrectly justified

### Definition An infinite sequence of numbers is an ordered set of real numbers.

Ifiite sequeces (Sect. 0. Today s Lecture: Review: Ifiite sequeces. The Cotiuous Fuctio Theorem for sequeces. Usig L Hôpital s rule o sequeces. Table of useful its. Bouded ad mootoic sequeces. Previous

### 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 + =

### 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

### NATIONAL UNIVERSITY OF SINGAPORE FACULTY OF SCIENCE SEMESTER 1 EXAMINATION ADVANCED CALCULUS II. November 2003 Time allowed :

NATIONAL UNIVERSITY OF SINGAPORE FACULTY OF SCIENCE SEMESTER EXAMINATION 003-004 MA08 ADVANCED CALCULUS II November 003 Time allowed : hours INSTRUCTIONS TO CANDIDATES This examiatio paper cosists of TWO

### 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

### 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

### Please do NOT write in this box. Multiple Choice. Total

Istructor: Math 0560, Worksheet Alteratig Series Jauary, 3000 For realistic exam practice solve these problems without lookig at your book ad without usig a calculator. Multiple choice questios should

### Notice that this test does not say anything about divergence of an alternating series.

MATH 572H Sprig 20 Worksheet 7 Topics: absolute ad coditioal covergece; power series. Defiitio. A series b is called absolutely coverget if the series b is coverget. If the series b coverges, while b diverges,

### Lesson 10: Limits and Continuity

www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

### Math 19B Final. Study Aid. June 6, 2011

Math 9B Fial Study Aid Jue 6, 20 Geeral advise. Get plety of practice. There s a lot of material i this sectio - try to do as may examples ad as much of the homework as possible to get some practice. Just

### MATH 312 Midterm I(Spring 2015)

MATH 3 Midterm I(Sprig 05) Istructor: Xiaowei Wag Feb 3rd, :30pm-3:50pm, 05 Problem (0 poits). Test for covergece:.. 3.. p, p 0. (coverges for p < ad diverges for p by ratio test.). ( coverges, sice (log

### Roberto s Notes on Series Chapter 2: Convergence tests Section 7. Alternating series

Roberto s Notes o Series Chapter 2: Covergece tests Sectio 7 Alteratig series What you eed to kow already: All basic covergece tests for evetually positive series. What you ca lear here: A test for series

### Solutions to Homework 7

Solutios to Homework 7 Due Wedesday, August 4, 004. Chapter 4.1) 3, 4, 9, 0, 7, 30. Chapter 4.) 4, 9, 10, 11, 1. Chapter 4.1. Solutio to problem 3. The sum has the form a 1 a + a 3 with a k = 1/k. Sice

### INFINITE SERIES PROBLEMS-SOLUTIONS. 3 n and 1. converges by the Comparison Test. and. ( 8 ) 2 n. 4 n + 2. n n = 4 lim 1

MAC 3 ) Note that 6 3 + INFINITE SERIES PROBLEMS-SOLUTIONS 6 3 +

### 10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term.

0. Sequeces A sequece is a list of umbers writte i a defiite order: a, a,, a, a is called the first term, a is the secod term, ad i geeral eclusively with ifiite sequeces ad so each term Notatio: the sequece

### Section 1.4. Power Series

Sectio.4. Power Series De itio. The fuctio de ed by f (x) (x a) () c 0 + c (x a) + c 2 (x a) 2 + c (x a) + ::: is called a power series cetered at x a with coe ciet sequece f g :The domai of this fuctio

### (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

### Seunghee Ye Ma 8: Week 5 Oct 28

Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value

### 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

### n n 2 n n + 1 +

Istructor: Marius Ioescu 1. Let a =. (5pts) (a) Prove that for every ε > 0 there is N 1 such that a +1 a < ε if N. Solutio: Let ε > 0. The a +1 a < ε is equivalet with + 1 < ε. Simplifyig, this iequality

### Part I: Covers Sequence through Series Comparison Tests

Part I: Covers Sequece through Series Compariso Tests. Give a example of each of the followig: (a) A geometric sequece: (b) A alteratig sequece: (c) A sequece that is bouded, but ot coverget: (d) A sequece

### Math 116 Practice for Exam 3

Math 6 Practice for Eam 3 Geerated April 4, 26 Name: SOLUTIONS Istructor: Sectio Number:. This eam has questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur

### MTH 142 Exam 3 Spr 2011 Practice Problem Solutions 1

MTH 42 Exam 3 Spr 20 Practice Problem Solutios No calculators will be permitted at the exam. 3. A pig-pog ball is lauched straight up, rises to a height of 5 feet, the falls back to the lauch poit ad bouces

### 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

### 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

### Testing for Convergence

9.5 Testig for Covergece Remember: The Ratio Test: lim + If a is a series with positive terms ad the: The series coverges if L . The test is icoclusive if L =. a a = L This

### THE INTEGRAL TEST AND ESTIMATES OF SUMS

THE INTEGRAL TEST AND ESTIMATES OF SUMS. Itroductio Determiig the exact sum of a series is i geeral ot a easy task. I the case of the geometric series ad the telescoig series it was ossible to fid a simle