Multivariable calculus  concepts and contexts 4th ed

Fourth Editio n R eview ers Jennifer Bailey, Colorado School o f Mines Lewis Blake, Duke University James Cook, North Carolina State U niversity Costel Ionita.. Dixie State College Lawre

Cengage has dramatically enhanced the online experience Interactive activities, animations, exercises, and topic-based lecture videos have been added to

Stewart’s Calculus: Concepts & Contexts, 4e, WebAssign component

Tai ngay!!! Ban co the xoa dong chu nay!!!

Multivariable Calculus

Concepts and Contexts | 4e

Calculus and the Architecture of Curves

T h e c o ver photograph s h o w s the

DZ B an k in B e rlin , design ed and

b u ilt 1995-2001 by Frank G e h ry

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th at Frank G e h ry d e sig n s w o u ld be

im p o ssib le to build w ith o u t the com p uter.

T h e C A T IA so ftw are that h is a rch i­

te cts and e n g in eers use to p ro duce the

co m p u te r m o dels is based on p rin cip le s of

c a lc u lu s —fitting cu rve s by m atching tangent

lin e s , m aking sure the cu rvatu re isn 't too

la rg e , and contro lling p aram etric su rfa ce s.

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of fre e d o m I can play w ith shapes."

T h e p ro ce ss starts w ith G e h ry's initial

sketches, w hich are translated into a su c ce s­

sio n of p h ysical m o d els (H und reds of different

p h ysical m o d els w e re constructed during the design

of the b uild ing, firs t w ith basic w oo den blocks and then

e vo lvin g into m ore sculp tural fo rm s )T h e n an engineer

u se s a digitizer to record the co o rd in ates of a se rie s of

p o ints on a physical m o d e l.T h e digitized points are fed

into a com p uter and the C A TIA so ftw are is used to link

th e se points w ith sm ooth cu rve s (It jo in s cu rve s so that

th e ir tangent lin es coincide; you can use the sam e idea to

design the sh a p e s of letters in the Labo rato ry Project on

page 208 of th is b o o k.)T h e architect has considerab le fre e ­

dom in creating these cu rve s, guided by d isp lays of the

cu rve , its d e rivative , and its c u rva tu re T h e n the cu rve s are

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co n n ected to each o ther by a p aram e tric su rfa c e ,

and ag ain the arch itect can do so in m an y p o ssib le

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a n o th e r p h ysica l m o d el, w h ic h , in tu rn , su g g e s ts

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by D a ssa u it S y s tè m e s , o rig in a lly fo r d esig n in g

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h is c o m p le x sc u lp tu ra l s h a p e s , is the firs t to use

it in a rch ite ctu re It h e lp s him a n s w e r his q u e s ­ tio n , "H o w w ig g ly can yo u get and still m ake a

b u ild in g ?"

Multivariable Calculus: Concepts and Contexts,

Fourth Edition Enhanced Edition

P h o to R esea rche r: L u m ina D a ta m a tic s

T e xt R e sea rche r: Lu m in a D a ta m a tic s

C o p y E d ito r: K a th i Tow nes

Illustrator: Brian Betsill

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8.3 The Integral and Comparison Tests: Estimating Sums 575

8.6 Representations of Functions as Power Series 598

Laboratory Project ■ An Elusive Limit 618 Writing Project ■ How Newton Discovered the Binomial Series 618

Applied Project ■ Radiation from the Stars 627

Focus on Problem Solving 631

Discovery Project ■ The Geometry of a Tetrahedron 662

9.5 Equations of Lines and Planes 663

Laboratory Project ■ Putting 3D in Perspective 672

9.7 Cylindrical and Spherical Coordinates 682

Laboratory Project ■ Families of Surfaces 687

Focus on Problem Solving 691

J

V I I

viii

10.2 Derivatives and Integrals of Vector Functions 701

10.4 Motion in Space: Velocity and Acceleration 716

Applied Project ■ Kepler's Laws 726

10.5 Parametric Surfaces 727

Focus on Problem Solving 735

11.1 Functions of Several Variables 738

11.2 Limits and Continuity 749

11.3 Partial Derivatives 756

11.4 Tangent Planes and Linear Approximations 770

11.6 Directional Derivatives and the Gradient Vector 789

Applied Project ■ Designing a Dumpster 811 Discovery Project ■ Quadratic Approximations and Critical Points 812

12.3 Double Integrals over General Regions 844

12.4 Double Integrals in Polar Coordinates 853

12.5 Applications of Double Integrals 858

12.9 Change of Variables in Multiple Integrals 891

Writing Project ■ Three Men and Two Theorems 966

13.8 The Divergence Theorem 967

Why is this text referred to as an “Enhanced Edition”?

Though the content of the text itself is essentially unchanged—it's still the original text authored by the late James Stewart—Cengage has dramatically enhanced the online expe­rience Interactive activities, animations, exercises, and new topic-based lecture videos have been added to WebAssign The text is now available in our “digital first” version, which includes a loose-leaf copy of the text and a printed WebAssign access code in a nicely priced bundle

INSTRUCTORS: For more information please contact your Cengage Learning Consultant who will be able to provide details on the latest online innovations and the

various purchase options for Calculus: Concepts & Contexts, 4e Enhanced Edition.

When the first edition of this book appeared twelve years ago, a heated debate about cal­culus reform was taking place Such issues as the use of technology, the relevance of rigor, and the role of discovery versus that of drill were causing deep splits in mathematics departments Since then the rhetoric has calmed down somewhat as reformers and tradi­tionalists have realized that they have a common goal: to enable students to understand and appreciate calculus

The first three editions were intended to be a synthesis of reform and traditional approaches to calculus instruction In this fourth edition I continue to follow that path by

emphasizing conceptual understanding through visual, verbal, numerical, and algebraic

approaches I aim to convey to the student both the practical power of calculus and the intrinsic beauty of the subject

What's New In the Fourth Edition?

The changes have resulted from talking with my colleagues and students at the University

of Toronto and from reading journals, as well as suggestions from users and reviewers Here are some of the many improvements that I've incorporated into this edition:

■ The majority of examples now have titles

■ Some material has been rewritten for greater clarity or for better motivation See, for instance, the introduction to series on page 565

■ New examples have been added and the solutions to some of the existing examples have been amplified

■ A number of pieces of art have been redrawn

■ The data in examples and exercises have been updated to be more timely

■ Sections 8.7 and 8.8 have been merged into a single section 1 had previously featured the binomial series in its own section to emphasize its importance But

I learned that some instructors were omitting that section, so I decided to incor­porate binomial series into 8.7

■ More than 25% of the exercises in each chapter are new Here are a few of my favorites: 8.2.35,9.1.42, 11.1.10-11, 11.6.37-38 11.8.20-21 and 13.3.21-22

■ There are also some good new problems in the Focus on Problem Solving sections See, for instance, Problem 13 on page 632, Problem 8 on page 692, Problem 9 on page 736, and Problem 11 on page 904

X I

The most important way to foster conceptual understanding is through the problems that

we assign To that end I have devised various types of problems Some exercise sets begin with requests to explain the meanings of the basic concepts of the section (See, for instance, the first couple of exercises in Sections 8.2, 11.2, and 11.3 I often use them as a basis for classroom discussions.) Similarly, review sections begin with a Concept Check and a True-False Quiz Other exercises test conceptual understanding through graphs or tables (see Exercises 8.7.2, 10.2.1-2, 10.3.33-37,11.1.1-2, 11.1.9-18, 11.3.3-10, 11.6.1-2, 11.7.3-4, 12.1.5-10, 13.1.11-18, 13.2.15-16, and 13.3.1-2)

Each exercise set is carefully graded, progressing from basic conceptual exercises and skill- development problems to more challenging problems involving applications and proofs

My assistants and I have spent a great deal of time looking in libraries, contacting compa­nies and government agencies, and searching the Internet for interesting real-world data

to introduce, motivate, and illustrate the concepts of calculus As a result, many of the examples and exercises deal with functions defined by such numerical data or graphs See, for instance, Example 3 in Section 9.6 (wave heights)

Functions of two variables are illustrated by a table of values of the wind-chill index as

a function of air temperature and wind speed (Example 1 in Section 11.1) Partial deriva­tives are introduced in Section 11.3 by examining a column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity This example is pursued further in connection with linear approximations (Example 3 in Section 11.4) Directional derivatives are introduced in Section 11.6 by using a temperature contour map to estimate the rate of change of temperature at Reno

in the direction of Las Vegas Double integrals are used to estimate the average snowfall

in Colorado on December 20-21, 2006 (Example 4 in Section 12.1) Vector fields are introduced in Section 13.1 by depictions of actual velocity vector fields showing San Fran­cisco Bay wind patterns

One way of involving students and making them active learners is to have them work (per­haps in groups) on extended projects that give a feeling of substantial accomplishment when completed A pplied Projects involve applications that are designed to appeal to the imagi­nation of students The project after Section 11.8 uses Lagrange multipliers to determine the masses of the three stages of a rocket so as to minimize the total mass while enabling the rocket to reach a desired velocity D iscovery Projects explore aspects of geometry: tetrahe- dra (after Section 9.4), hyperspheres (after Section 12.7), and intersections of three cylin­ders (after Section 12.8) The Laboratory Project on page 687 uses technology to discover how interesting the shapes of surfaces can be and how these shapes evolve as the parame­ters change in a family The Writing Project on page 966 explores the historical and physi­cal origins of Green’s Theorem and Stokes’ Theorem and the interactions of the three men involved Many additional projects are provided in the Instructor's Guide.

The availability of technology makes it not less important but more important to under­stand clearly the concepts that underlie the images on the screen But, when properly used, graphing calculators and computers are powerful tools for discovering and understanding those concepts I assume that the student has access to either a graphing calculator or a computer algebra system The icon ¡/g indicates an exercise that definitely requires the use

of such technology, but that is not to say that a graphing device can’t be used on the other exercises as well The symbol [t] is reserved for problems in which the full technological resources of Derive, Maple, Mathematica, or the TI-89/92 are required But technology doesn’t make pencil and paper obsolete Hand calculation and sketches are often prefer­able to technology for illustrating and reinforcing some concepts Both instructors and stu­dents need to develop the ability to decide where the hand or the machine is appropriate

PREFACE xiii

Homework Hints

WebAssign

W ebsite: w w w stew artcalcu lu s.co m

Homework Hints are representative exercises (usually odd-numbered) in every section of

the text, indicated by printing the exercise number in red These hints are usually present­

ed in the form of questions and try to imitate an effective teaching assistant by function­ing as a silent tutor They are constructed so as not to reveal any more of the actual solution than is minimally necessary to make further progress

* V WEBASSIGN www.webassign.com/cengage

t % From Cengigc

WebAssign from Cengage Calculus: Concepts & Contexts 4e Enhanced Edition is a fully

customizable online solution, including an interactive ebook, for STEM disciplines that empowers you to help your students learn, not just do homework Insightful tools save you time and highlight exactly where your students are struggling Decide when and what type of help students can access while working on assignments— and incentivize inde­pendent work so help features aren’t abused Meanwhile, your students get an engaging experience, instant feedback and better outcomes A total win-win!

To try a sample assignment, learn about LMS integration or connect with our digital course support visit www.webassign.com/cengage

This website includes the following

■ Algebra Review

■ Lies My Calculator and Computer Told Me

■ History of Mathematics, with links to the better historical websites

■ Additional Topics (complete with exercise sets):

Trigonometric Integrals, Trigonometric Substitution, Strategy for Integration, Strategy for Testing Series, Fourier Series, Formulas for the Remainder Term in Taylor Series, Linear Differential Equations, Second-Order Linear Differential Equations, Nonhomogeneous Linear Equations, Applications of Second-Order Differential Equations, Using Series to Solve Differential Equations, Rotation

of Axes, and (for instructors only) Hyperbolic Functions

■ Links, for each chapter, to outside Web resources

■ Archived Problems (drill exercises that appeared in previous editions, together with their solutions)

■ Challenge Problems (some from the Focus on Problem Solving sections of prior editions)

Content

8 ■ Infinite Seq uen ces and Series

Tests for the convergence of series are considered briefly, with intuitive rather than for­mal justifications Numerical estimates of sums of series are based on which lest was used to prove convergence The emphasis is on Taylor series and polynomials and their applications to physics Error estimates include those from graphing devices

9 ■ Vectors and The G eom etry of Sp ace

The dot product and cross product of vectors are given geometric definitions, motivated by work and torque, before the algebraic expressions are deduced To facilitate the discussion

of surfaces, functions of two variables and their graphs are introduced here

10 ■ Vector Functions

The calculus of vector functions is used to prove Kepler's First Law of planetary motion, with the proofs of the other laws left as a project In keeping with the introduction of para­metric curves in Chapter l, parametric surfaces are introduced as soon as possible, namely

in this chapter I think an early familiarity with such surfaces is desirable, especially with the capability of computers to produce their graphs Then tangent planes and areas of para­metric surfaces can be discussed in Sections 11.4 and 12.6.

11 ■ Partial Derivatives

Functions of two or more variables are studied from verbal, numerical, visual, and alge­braic points of view In particular, I introduce partial derivatives by looking at a specific column in a table of values of the heat index (perceived air temperature) as a function of the actual temperature and the relative humidity Directional derivatives are estimated from contour maps of temperature, pressure, and snowfall

12 ■ Multiple Integrals

Contour maps and the Midpoint Rule are used to estimate the average snowfall and average temperature in given regions Double and triple integrals are used to compute probabil­ities, areas of parametric surfaces, volumes of hyperspheres, and the volume of intersection

of three cylinders

13 ■ Vector Fields

Vector fields are introduced through pictures of velocity fields showing San Francisco Bay wind patterns The similarities among the Fundamental Theorem for line integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem are emphasized

Ancillaries

plete set of ancillaries developed under my direction Each piece has been designed to enhance student understanding and to facilitate creative instruction The table on pages xviii and xix lists ancillaries available for instructors and students

Acknowledgments

I am grateful to the following reviewers for sharing their knowledge and judgment with

me I have learned something from each of them

Fourth Editio n R eview ers

Jennifer Bailey, Colorado School o f Mines

Lewis Blake, Duke University

James Cook, North Carolina State U niversity

Costel Ionita Dixie State College

Lawrence Levine, Stevens Institute o f Technology

Scott Mortensen, Dixie State College

Drew Pasteur, North Carolina State University

Jeffrey Powell, S an ford University

Barbara Tozzi, Brookdale Community College

Kathryn Turner, Utah State University

Cathy Zucco-Tevelof, Arcadia University

Previous Edition R eview ers

Irfan Altas Charles Sturt University

William Ardis, Collin County Community College

Barbara Bath, Colorado School o f Mines

Neil Berger, University o f Illinois at Chicago

Jean H Bevis Georgia State University

Martina Bode, Northwestern University

Jay Bourland Colorado State University

Paul Wayne Britt, Louisiana State University

Judith Broad win, Jericho High School (retired)

Charles Bu, Wellesley University

Meghan Anne Burke, Kennesaw State University

Robert Burton, Oregon State University

Roxanne M Byrne, University o f Colorado at Denver

Maria E Calzada, Loyola U niversity-N ew Orleans

Larry Cannon, Utah State University

Deborah Troutman Cantrell,

Chattanooga State Technical Community College

Bern Cayco, San Jose State University

PREFACE xv

John Chadam, University o f Pittsburgh

Robert A Chaffer, Central Michigan University

Dan Clegg, Palomar College

Camille P Cochrane, Shelton State Community College

James Daly, University o f Colorado

Richard Davis, Edmonds Community College

Susan Dean, DeAnza College

Richard DiDio, LaSalle University

Robert Dieffenbach, Miami University-Middletown

Fred Dodd, University o f South Alabama

Helmut Doll, Bloomsburg University

William Dunham, Muhlenberg College

David A Edwards, The University o f Georgia

John Ellison, Grove City College

Joseph R Fiedler, California State University-Bakersfield

Barbara R Fink, DeAnza College

James P Fink, Gettysburg College

Joe W Fisher, University o f Cincinnati

Robert Fontenot, Whitman College

Richard L Ford, California State University Chico

Laurette Foster, Prairie View A & M University

Ronald C Freiwald, Washington University in St Louis

Frederick Gass, Miami University

Gregory Goodhart, Columbus State Community College

John Gosselin, University o f Georgia

Daniel Grayson,

University o f Illinois at Urbana-Champaign

Raymond Green well, Hofstra University

Gerrald Gustave Greivel, Colorado School oj Mines

John R Griggs, North Carolina State University

Barbara Bell Grover, Salt Lake Community College

Murli Gupta, The George Washington University

John William Hagood, Northern Arizona University

Kathy Hann, California State University at Hayward

Richard Hitt, University o f South Alabama

Judy Holdener, United States Air Force Academy

Randall R Holmes, Auburn University

Barry D Hughes, University o f Melbourne

Mike Hurley, Case Western Reserve University

Gary Steven Itzkowitz, Rowan University

Helmer Junghans, Montgomery College

Victor Kaftal, University o f Cincinnati

Steve Kahn, Anne Arundel Community College

Mohammad A Kazemi,

University o f North Carolina, Charlotte Harvey Keynes, University o f Minnesota Kandace Alyson Kling, Portland Community College Ronald Knill, Tulane University

Stephen Kokoska, Bloomsburg University Kevin Kreider, University o f Akron Doug Kuhlmann, Phillips Academy David E Kullman, Miami University Carrie L Kyser, Clackamas Community College Prem K Kythe, University o f New Orleans James Lang, Valencia Community College—East Campus Carl Leinbach, Gettysburg College

William L Lepowsky, Laney College Kathryn Lesh, University o f Toledo Estela Llinas, University o f Pittsburgh at Greensbnrg

Beth Turner Long,

Pellissippi State Technical Community College Miroslav Lovric, McMaster University

Lou Ann Mahaney, Tarrant County Junior College-Northeast John R Martin, Tarrant County Junior College

Andre Mathurin, Bellarmine College Prep

R J McKellar, University o f New Brunswick

Jim McKinney,

California State Polytechnic University-Pomona Richard Eugene Mercer, Wright State University David Minda, University o f Cincinnati

Rennie Mirollo, Boston College Laura J Moore-Mueller, Green River Community College Scott L Mortensen, Dixie State College

Brian Mortimer, Carleton University Bill Moss, Clemson University

Tejinder Singh Neelon,

California State University San Marcos Phil Novinger, Florida State University Richard Nowakowski, Dalhousie University Stephen Ott, Lexington Community College Grace Orzech, Queen \s University

Jeanette R Palmiter, Portland State University' Bill Paschke, University o f Kansas

David Patocka Tulsa Community C ollege— Southeast Campus Paul Patten, North Georgia College

Leslie Peek Mercer University

Mike Pepe, Seattle Central Community College

Dan Pritikin, Miami University

Fred Prydz, Shoreline Community College

Denise Taunton Reid, Valdosta State University

James Reynolds, Clarion University

Heman Rivera, Texas Lutheran University

Richard Rochberg, Washington University

Gil Rodriguez, Los Medaños College

David C Royster, University o f North Carolina-Charlotte

Daniel Russow, Arizona Western College

Dusty Edward Sabo, Southern Oregon University

Daniel S Sage, Louisiana State University

N Paul Schembari, East Stroudsburg University

Dr John Schmeelk, Virginia Commonwealth University

Bettina Schmidt Auburn University at Montgomery

Bemd S.W Schroeder, Louisiana Tech University

Jeffrey Scott Scroggs, North Carolina State University

James F Selgrade, North Carolina State University

Brad Shelton, University o f Oregon

Don Small, United States Military Academy-West Point Linda E Sundbye, The Metropolitan State College o f Denver Richard B Thompson, The University o f Arizona

William K Tomhave, Concordia College Lorenzo Traldi, Lafayette College Alan Tucker, State University o f New York at Stony Brook Tom Tucker, Colgate University

George Van Zwalenberg, Calvin College Dennis Watson, Clark College

Paul R Wenston, The University o f Georgia Ruth Williams, University o f Califomia-San Diego Clifton Wingard, Delta State University

Jianzhong Wang, Sam Houston State University JingLing Wang, Lansing Community College Michael B Ward, Western Oregon University Stanley Wayment, Southwest Texas State University Barak Weiss, Ben Gurion University-Be’er Sheva, Israel Teri E Woodington, Colorado School o f Mines

James Wright, Keuka College

In addition, I would like to thank Ari Brodsky, David Cusick, Alfonso Gracia-Saz, Emile LeBlanc, Tanya Leise, Joe May, Romaric Pujol, Norton Starr, Lou Talman, and Gail Wolkowicz for their advice and suggestions; A1 Shenk and Dennis Zill for permission to use exercises from their calculus texts; COMAP for permission to use project material; Alfonso Gracia-Saz, B Hovinen, Y Kim, Anthony Lam, Romaric Pujol, Felix Recio, and Paul Sally for ideas for exercises; Dan Drucker for the roller derby project; and Tom Farmer, Fred Gass, John Ramsay, Larry Riddle, V.K Srinivasan, and Philip Straffin for ideas for projects I’m grateful to Dan Clegg, Jeff Cole, and Tim Flaherty for preparing the answer manuscript and suggesting ways to improve the exercises

As well, 1 thank those who have contributed to past editions: Ed Barbeau, George Bergman, David Bleecker, Fred Brauer, Andy Bulman-Fleming, Tom DiCiccio, Martin Erickson, Garret Etgen, Chris Fisher, Stuart Goldenberg, Arnold Good, John Hagood, Gene Hecht, Victor Kaftal, Harvey Keynes, E L Koh, Zdislav Kovarik, Kevin Kreider, Jamie Lawson, David Leep, Gerald Leibowitz, Larry Peterson, Lothar Redlin, Peter Rosen­thal, Carl Riehm, Ira Rosenholtz, Doug Shaw, Dan Silver, Lowell Smylie, Larry Wallen, Saleem Watson, and Alan Weinstein

I also thank Stephanie Kuhns, Rebekah Million, Brian Betsill, and Kathi Townes of TECH-arts for their production services; Marv Riedesel and Mary Johnson for their care­ful proofing of the pages; Thomas Mayer for the cover image; and the following Brooks/ Cole staff: Cheryll Linthicum, editorial production project manager; Jennifer Jones, Angela Kim, and Mary Anne Payumo, marketing team; Peter Galuardi, media editor; Jay Campbell, senior developmental editor; Jeannine Lawless, associate editor; Elizabeth Neustaetter, editorial assistant; Bob Kauser, permissions editor; Becky Cross, print/media buyer; Vernon Boes, art director; Rob Hugel, creative director; and Irene Morris, cover designer They have all done an outstanding job.

I have been very fortunate to have worked with some of the best mathematics editors in the business over the past three decades: Ron Munro, Harry Campbell, Craig Barth, Jeremy Hay- hurst, Gary Ostedt, Bob Pirtle, and now Richard Stratton Special thanks go to all of them

JAM ES STEW ART

Ancillaries for Instructors

PowerLecture CD-ROM with Joinln and ExamView

IS B N 978-0-495-56049-4

Contains all art from the text in both jpeg and PowerPoint

formats, key equations and tables from the text, complete

pre-built PowerPoint lectures, and an electronic version o f

the Instructor's Guide Also contains Joinln on TumingPoint

personal response system questions and ExamView algor­

ithmic test generation See below fo r complete descriptions.

Instructor's Guide

by Douglas Shaw and James Stewart

IS B N 978-0-495-56047-0

Each section of the main text is discussed from several view­

points and contains suggested time to allot, points to stress,

text discussion topics, core materials for lecture, workshop/

discussion suggestions, group work exercises in a form suit­

able for handout, and suggested homework problems An elec­

tronic version is available on the PowerLecture CD-ROM.

Instructor's Guide for AP® Calculus

by Douglas Shaw

IS B N 978-0-495-56059-3

Taking the perspective of optimizing preparation for the AP

exam, each section of the main text is discussed from several

viewpoints and contains suggested time to allot, points to

stress, daily quizzes, core materials for lecture, workshop/

discussion suggestions, group work exercises in a form suit­

able for handout, tips for the AP exam, and suggested home­

work problems.

Complete Solutions Manual, Multivariable

by Dan Clegg

IS B N 978-0-495-56056-2

Includes worked-out solutions to all exercises in the text.

Printed Test Bank

by William Tomhave and Xuequi Zeng

IS B N 978-0-495-56123-1

Contains multiple-choice and short-answer test items that key

directly to the text.

Create, deliver, and customize tests and study guides (both print and online) in minutes with this easy-to-use assessment and tutorial software on CD Includes full algorithmic genera­ tion o f problems and complete questions from the Printed Test Bank.

Text-Specific DVDs

IS B N 978-0-495-56050-0

Text-specific DVD set, available at no charge to adopters Each disk features a 10- to 20-minute problem-solving lesson for each section of the chapter Covers both single- and multi- variable calculus.

Ancillaries for Instructors and Students

Stewart Specialty Website

www.stewartcalculus.com

Contents: Algebra Review ■ Additional Topics ■ Drill exercises ■ Challenge Problems ■ Web Links ■ History of Mathematics

WEBASSIGN www.webassign.com

Prepare fo r class with confidence using WebAssign from

This online learning platform, which includes an interactive ebook, fuels practice, so you truly absorb what you learn—and are better prepared come test time Videos and tutorials walk you through concepts and deliver instant feedback and grading,

so you always know where you stand in class Focus your study time and get extra practice where you need it most Study smarter with WebAssign!

Ask your instructor today how you can get access to WebAssign, or learn about self-study options at www.webassign.com

The Mathematics Resource Center Website

www.cengage.com/math

When you adopt a Cengage Learning mathematics text, you and your students will have access to a variety of teaching and learning resources This website features everything from book- specific resources to newsgroups It’s a great way to make teaching and learning an interactive and intriguing experience.

Student Resources

Study Guide, Multivariable

by Robert Burton and Dennis Garity

IS B N 978-0-495-56057-9

Contains key concepts, skills to master, a brief discussion of

the ideas o f the section, and worked-out examples with tips

on how to find the solution.

Student Solutions Manual, Multivariable

by Dan Clegg

IS B N 978-0-495-56055-5

Provides completely worked-out solutions to all odd-numbered

exercises within the text, giving students a way to check their

answers and ensure that they took the correct steps to arrive

at an answer.

CalcLabs with Maple, Multivariable

by Philip B Yasskin and Art Belmonte

IS B N 978-0-495-56058-6

CalcLabs with Mathematica, Multivariable

by Selwyn Hollis

IS B N 978-0-495-82722-1

Each o f these comprehensive lab manuals will help students

learn to effectively use the technology tools available to them

Each lab contains clearly explained exercises and a variety of

labs and projects to accompany the text.

A Companion to Calculus, Second Edition

by Dennis Ebersole, Doris Schattschneider, Alicia Sevilla, and Kay Somers

IS B N 978-0-495-01124-8

Written to improve algebra and problem-solving skills of students taking a calculus course, every chapter in this com­ panion is keyed to a calculus topic, providing conceptual background and specific algebra techniques needed to under­ stand and solve calculus problems related to that topic It is designed fo r calculus courses that integrate the review of pre­ calculus concepts or fo r individual use.

Linear Algebra for Calculus

by Konrad J Heuvers, William P Francis, John H Kuisti, Deborah F Lockhart, Daniel S Moak, and Gene M Ortner

IS B N 978-0-534-25248-9

This comprehensive book, designed to supplement the calculus course, provides an introduction to and review o f the basic ideas o f linear algebra.

To the Student

Reading a calculus textbook is different from reading a news­

paper or a novel, or even a physics book Don’t be discouraged

if you have to read a passage more than once in order to under­

stand it You should have pencil and paper and calculator at

hand to sketch a diagram or make a calculation

Some students start by trying their homework problems and

read the text only if they get stuck on an exercise I suggest that

a far better plan is to read and understand a section of the text

before attempting the exercises In particular, you should look

at the definitions to see the exact meanings of the terms And

before you read each example, I suggest that you cover up the

solution and try solving the problem yourself You’ll get a lot

more from looking at the solution if you do so

Part of the aim of this course is to train you to think logically

Learn to write the solutions of the exercises in a connected,

step-by-step fashion with explanatory sentences— not just a

string of disconnected equations or formulas

The answers to the odd-numbered exercises appear at the

back of the book, in Appendix J Some exercises ask for a ver­

bal explanation or interpretation or description In such cases

there is no single correct way of expressing the answer, so don’t

worry that you haven’t found the definitive answer In addition,

there are often several different forms in which to express a

numerical or algebraic answer, so if your answer differs from

mine, don’t immediately assume you’re wrong For example,

if the answer given in the back of the book is y/2 — 1 and you

obtain l/(l + yfl), then you’re right and rationalizing the

denominator will show that the answers are equivalent

The icon H indicates an exercise that definitely requires the use of either a graphing calculator or a computer with graphing software (Section 1.4 discusses the use of these graphing devices and some of the pitfalls that you may encounter.) But that doesn’t mean that graphing devices can’t

be used to check your work on the other exercises as well.The symbol [t] is reserved for problems in which the full technological resources of Derive, Maple, Mathematica, or the TI-89/92 are required

You will also encounter the symbol which warns you against committing an error I have placed this symbol in the margin in situations where I have observed that a large propor­tion of my students tend to make the same mistake

I recommend that you keep this book for reference purposes after you finish the course Because you will likely forget some

of the specific details of calculus, the book will serve as a useful reminder when you need to use calculus in subsequent courses And, because this book contains more material than can be covered in any one course, it can also serve as a valuable resource for a working scientist or engineer

Calculus is an exciting subject, justly considered to be one

of the greatest achievements of the human intellect I hope you will discover that it is not only useful but also intrinsically beautiful

JAMES STEWART

X X

Infinite sequences and series were introduced briefly in /\ Preview of Calculus

in connection with Zeno's paradoxes and the decimal representation of numbers Their importance in calculus stems from Newton's idea of representing functions

as sums of infinite series For instance, in finding areas he often integrated a function by first expressing it as a series and then integrating each term of the series We will pursue his idea in Section 8.7 in order to integrate such functions

as e~x\ (Recall that we have previously been unable to do this.) Many of the

functions that arise in mathematical physics and chemistry, such as Bessel func­tions, are defined as sums of series, so it is important to be familiar with the basic concepts of convergence of infinite sequences and series

Physicists also use series in another way, as we will see in Section 8.8 In studying fields as diverse as optics, special relativity, and electromagnetism, they analyze phenomena by replacing a function with the first few terms in the series that represents it

Notice that for every positive integer n there is a corresponding number a„ and so a

sequence can be defined as a function whose domain is the set of positive integers But we

usually write an instead of the function notation f{n) for the value of the function at the number n.

Notation: The sequence {ai , a2, ai, } is also denoted by

EXAMPLE 1 Describing sequences Some sequences can be defined by giving a formula

for the nth term In the following examples we give three descriptions of the sequence:

one by using the preceding notation, another by using the defining formula, and a third

by writing out the terms of the sequence Notice that n doesn’t have to start at 1.

(a)

n + 1 n

(b)

(c) {V« _ 3 }*_3 a„ = V« - 3, n 3* 3 {O, 1, \/2 , V T , V« - 3 ,

(d)

Q EXAMPLE 2 Find a formula for the general term a„ of the sequence

assuming that the pattern of the first few terms continues

Notice that the numerators of these fractions start with 3 and increase by 1 whenever we

go to the next term The second term has numerator 4, the third term has numerator 5; in general, the nth term will have numerator n + 2 The denominators are the powers of 5,

SECTION 8.1 SEQUENCES 555

so an has denominator 5" The signs of the terms are alternately positive and negative,

so we need to multiply by a power of - 1 In Example 1(b) the factor ( - 1 ) " meant we started with a negative term Here we want to start with a positive term and so we use (—l ) n_1 or (— l ) rt+1 Therefore

an = ( - i y - ' n + 2

5 n

EXAMPLE 3 Here are some sequences that don’t have simple defining equations

(a) The sequence {pn}, where p n is the population of the world as of January 1 in the year n.

(b) If we let an be the digit in the rcth decimal place of the number e, then {<an} is a well-

defined sequence whose first few terms are

{7, 1,8, 2, 8, 1,8, 2, 8, 4, 5 , }

(c) The Fibonacci sequence {/„} is defined recursively by the conditions

/l = 1 / 2 = 1 f n = f n - l + f n - 2 3Each term is the sum of the two preceding terms The first few terms are

{1, 1,2, 3, 5, 8, 13,2 1 , }

This sequence arose when the 13th-century Italian mathematician known as Fibonacci

0

FIGURE 1

a4 A sequence such as the one in Example 1(a), a n = n/(n + 1), can be pictured either by ai<2^ i plotting its terms on a number line, as in Figure 1, or by plotting its graph, as in Figure 2

"7 ' : : ^ * Note that, since a sequence is a function whose domain is the set of positive integers, its

means that the terms of the sequence {an} approach L as n becomes large Notice that the

following definition of the limit of a sequence is very similar to the definition of a limit of

a function at infinity given in Section 2.5

[T] Definition A sequence {an} has the limit L and we write

lim an = L or an —> L as n —» oo

sequence is given in Appendix D.

if we can make the terms an as close to L as we like by taking n sufficiently large

If limn->* an exists, we say the sequence converges (or is convergent) Otherwise,

we say the sequence diverges (or is divergent).

Figure 3 illustrates Definition 1 by showing the graphs of two sequences that have the

If you compare Definition 1 with Definition 2.5.4 you will see that the only difference

between lim,,-,« an = L and lim *-,«/(jt) — L is that n is required to be an integer Thus

we have the following theorem, which is illustrated by Figure 4

|~2~| Theorem If lim*_«/(;c) = L and f(ri) = an when n is an integer, then

Limit Laws for Sequences

If {<an} and {bn} are convergent sequences and c is a constant, then

lim (an + bn) = lim an + lim bn

Another useful fact about limits of sequences is given by the following theorem, which

follows from the Squeeze Theorem because — | an | ^ an ^ | an |.

FIGURE 5

The sequence {bn} is squeezed

between the sequences {a n}

and {c„}.

This shows that the guess we made earlier

from Figures 1 and 2 was correct.

EXAMPLE 4 Find l i m -.

n + 1

SOLUTION The method is similar to the one we used in Section 2.5: Divide numerator

and denominator by the highest power of n that occurs in the denominator and then use

the Limit Laws

_ i: _ T _ " —x _

1lim 1 + lim —

EXAMPLE 5 Applying I'Hospital's Rule to a related function C a lculate lim

SOLUTION Notice that both numerator and denominator approach infinity as n —» *5 We can’t apply 1’Hospital’s Rule directly because it applies not to sequences but to functions

of a real variable However, we can apply 1’Hospital’s Rule to the related function

f ( x J = (In x)/x and obtain

I n x l / x

l i m - = l i m - = 0

x —*°° x 1Therefore, by Theorem 2, we have

Ill wlim - = 0

n-*oc n an

1

1

-FIGURE 6

EXAMPLE 6 Determine whether the sequence an = (— 1)” is convergent or divergent

SOLUTION If we write out the terms of the sequence, we obtain

The graph of this sequence is shown in Figure 6 Since the terms oscillate between 1 and

— 1 infinitely often, an does not approach any number Thus lim,,-*» (— 1)" does not exist;

The graph of the sequence in Example 7 is

shown in Figure 7 and supports the answer.

SOLUTION Because the sine function is continuous at 0, Theorem 5 enables us to write

\"-*x I

EJ EXAMPLE 9 Using the Squeeze Theorem D iscuss the convergence o f the sequence

an = n\/nn, w here n\ = 1 • 2 • 3 n.

have no corresponding function for use with 1’Hospital’s Rule (x\ is not defined when

x is not an integer) Let’s write out a few terms to get a feeling for what happens to an

SECTION 8.1 SEQUENCES 559

Creating Graphs of Sequences

Some computer algebra systems have special

commands that enable us to create sequences

and graph them directly With most graphing

calculators, however, sequences can be graphed

by using parametric equations For instance,

the sequence in Example 9 can be graphed by

entering the parametric equations

x = t y = t\/t'

and graphing in dot mode, starting with t = 1

and setting the r-step equal to 1 The result is

SOLUTION We know from Section 2.5 and the graphs of the exponential functions in

Section 1.5 that lim*—«, ax = °° for a > 1 and lim.v_>* a x = 0 for 0 < a < 1 Therefore, putting a = r and using Theorem 2, we have

i f A > l

[0 if 0 < r < 1

For the cases r = 1 and r = 0 we have

lim 1" = lim 1 = 1 and lim 0" = lim 0 = 0

The results of Example 10 are summarized for future use as follows.

|~T] The sequence {rn} is convergent if — 1 < r ^ 1 and divergent for all other

Definition A sequence {a,,} is called increasing if an < an+1 for all n ^ 1, that is,

cl \ < a2 < <23 < * * * It is called decreasing if an > an+] for all n ^ 1 A sequence

is monotonie if it is either increasing or decreasing

The right side is smaller because it has a

larger denominator.

n + 5 (n + 1) + 5 n + 6

and so an > a n+] for all n ^ 1

EXAMPLE 12 Show that the sequence an = , , ■ - is decreasing

Mathematical induction is often used in deal­

ing with recursive sequences See page 84 for

a discussion of the Principle of Mathematical

If it is bounded above and below, then {an} is a bounded sequence.

For instance, the sequence an = n is bounded below (an > 0) but not above The sequence an = n/(n + 1) is bounded because 0 < a„ < 1 for all n.

We know that not every bounded sequence is convergent [for instance, the sequence

an = (—1)" satisfies — 1 ^ a„ ^ 1 but is divergent, from Example 6] and not every mono­ tonic sequence is convergent (an = n —» oc) But if a sequence is both bounded and

monotonic, then it must be convergent This fact is stated without proof as Theorem 8, but

intuitively you can understand why it is true by looking at Figure 10 If {an} is increasing and an ^ M lor all //, then the terms are forced to crowd together and approach some num­ ber L.

[~8~| Monotonie Sequence Theorem Every bounded, monotonie sequence is convergent

EXAMPLE 13 The limit of a recursively defined sequence Investigate the sequence {an} defined by the recurrence relation

These initial terms suggest that the sequence is increasing and the terms are approaching

6 To confirm that the sequence is increasing, we use mathematical induction to show

that an+\ > an for all n ^ 1 This is true for n = 1 because a^ = 4 !> a i If we assume that it is true for n = k, then we have

Clk+ 1 > Uk

so a k , \ + 6 > ci k + 6

We have deduced that a n+1 > fl„ is true for n = k + 1 Therefore the inequality is true for all n by induction.

Next we verify that {an} is bounded by showing that an < 6 for all n (Since the sequence is increasing, we already know that it has a lower bound: an ^ a x = 2 for all n.) We know that a x < 6, so the assertion is true for n = 1 Suppose it is true for

11 Since the sequence {an} is increasing ana ’ y r i • • ^ o c i n a and bounded, the Monotonie Sequence Theoremdouhucu \

, V thonrem doesn t tell us what the value of the limit is.

But now that we know L = limn— a n exisii>,

1 (a) What is a sequence?

(b) What does it mean to say that lim,,-** an = 8?

(c) What does it mean to say that lim,,-»* a n = «>?

2 (a) What is a convergent sequence? Give two examples.

(b) What is a divergent sequence? Give two examples.

3 List the first six terms of the sequence defined by

n

Qn ~ 2n + 1

Does the sequence appear to have a limit? If so, find it.

4 List the first nine terms of the sequence {cos(n7r/3)| Does this

sequence appear to have a limit? If so, find it If not, explain

why.

5-10 Find a formula for the general term an of the sequence,

assuming that the pattern of the first few terms continues.

7 { 2 , 7 , 1 2 , 1 7 , } 8.

9 { 1 , - ! 2 4 9 »

10 {5, 1,5, 1,5, 1 , }

11-34 Determine whether the sequence converges or diverges

If it converges, find the limit.

I 35-40 Use a graph of the sequence to decide whether the

sequence is convergent or divergent If the sequence is conver­

gent, guess the value of the limit from the graph and then prove

your guess (See the margin note on page 559 for advice on

41 If $1000 is invested at 6% interest, compounded annually,

then after n years the investment is worth a„ = 1000(1.06)"

dollars.

(a) Find the first five terms of the sequence {a„}.

(b) Is the sequence convergent or divergent? Explain.

42 If you deposit $100 at the end of every month into an account

that pays 3% interest per year compounded monthly, the

amount of interest accumulated after n months is given by

the sequence

1.0025" - 1

In = 100 0.0025

(a) Find the first six terms of the sequence.

(b) How much interest will you have earned after two years?

43 A fish farmer has 5000 catfish in his pond The number of

catfish increases by 8% per month and the farmer harvests

300 catfish per month.

(a) Show that the catfish population P„ after n months is

given recursively by

(b) How many catfish are in the pond after six months?

44 Find the first 40 terms of the sequence defined by

and a\ = 11 Do the same if a\ = 25 Make a conjecture

about this type of sequence.

45 (a) Determine whether the sequence defined as follows is convergent or divergent:

(b) What happens if the first term is a x = 2 1

46 (a) If lim«-,« an = L, what is the value of lim,,.^* cin+]l (b) A sequence {an} is defined by

Find the first ten terms of the sequence correct to five decimal places Does it appear that the sequence is con­ vergent? If so, estimate the value of the limit to three decimal places.

Assuming that the sequence in part (b) has a limit, use part (a) to find its exact value Compare with your estimate from part (b).

Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age

2 months If we start with one newborn pair, how many pairs of rabbits will we have in the nth month? Show that

the answer is f n, where {/,} is the Fibonacci sequence

49—52 Determine whether the sequence is increasing, decreasing,

or not monotonic Is the sequence bounded?

53 Suppose you know that {an} is a decreasing sequence and

all its terms lie between the numbers 5 and 8 Explain why the sequence has a limit What can you say about the value

of the limit?

A sequence {an} is given by a x = x 2, a n+i — \ 2 + a,. (a) By induction or otherwise, show that {i/,t} is increasing and bounded above by 3 Apply the Monotonie Sequence Theorem to show that lim„ an exists.

(b) Find lim„ an.

55 Show that the sequence defined by

1

d 1— 1 <2n+l — j

an

is increasing and an < 3 for all n Deduce that {an} is conver­

gent and find its limit

56 Show that the sequence defined by

<21 2 £2/i+l ~1

3 — <2„

satisfies 0 < an ^ 2 and is decreasing Deduce that the

sequence is convergent and find its limit

57 We know that lim„_*x (0.8)" = 0 [from (7) with r = 0.8]

Use logarithms to determine how large n has to be so

that (0.8)” < 0.000001

positive constants that depend on the species and its environ­

ment Suppose that the population in year 0 is pQ> 0.

(a) Show that if {/?„} is convergent, then the only possible

values for its limit are 0 and b — a.

(b) Show that pn+i < (b/a)pn.

(c) Use part (b) to show that if a > b, then lim„—» pn — 0;

in other words, the population dies out

(d) Now assume that a < b Show that if p 0 < b — a, then {pn} is increasing and 0 < pn < b - a Show also that

if po > b — a, then {pn} is decreasing and pn > b — a Deduce that if a < b, then limn_**/?n = b - a.

60 A sequence is defined recursively by

a\ = 1 <2„+i = 1 + -

—; 1 + a n

58 (a) Let a, = a, a 2 = /(a), a 3 = / ( a 2) = /(/(* )),

limn-*x a„ = L, show that f(L) = L.

(b) Illustrate part (a) by taking f(x) = cos jc , a — 1, and

estimating the value of L to five decimal places.

59 The size of an undisturbed fish population has been modeled

by the formula

Pn+l bpn

a + pn where pn is the fish population after n years and a and b are

Find the first eight terms of the sequence {a„} What do you notice about the odd terms and the even terms? By considering the odd and even terms separately, show that {an} is convergent and deduce that

A sequence that arises in ecology as a model for population growth is defined by the logistic difference equation

An ecologist is interested in predicting the size of the population as time goes on, and asks these questions: Will it stabilize at a limiting value? Will it change in a cyclical fashion? Or will

it exhibit random behavior?

Write a program to compute the first n terms of this sequence starting with an initial popula­ tion po, where 0 < p0 < 1 Use this program to do the following.

1 Calculate 20 or 30 terms of the sequence for po = { and for two values of k such that

1 < k < 3 Graph each sequence Do the sequences appear to converge? Repeat for a dif­ ferent value of po between 0 and 1 Does the limit depend on the choice of p0? Does it depend on the choice of kl

2 Calculate terms of the sequence for a value of k between 3 and 3.4 and plot them What do

you notice about the behavior of the terms?

I f f ]

SECTION 8.2 SERIES 565

3. Experiment with values of k between 3.4 and 3.5 What happens to the terms?

4. For values of k between 3.6 and 4, compute and plot at least 100 terms and comment on the behavior of the sequence What happens if you change p0 by 0.001? This type of behavior is called chaotic and is exhibited by insect populations under certain conditions.

p la c e s b y S h ig e ru K o n d o a n d A le x a n d e r Y e e The convention behind our decimal notation is that any number can be written as an infi­

nite sum Here it means that

Does it make sense to talk about the sum of infinitely many terms?

It would be impossible to find a finite sum for the series

n Sum of first n terms

because if we start adding the terms we get the cumulative sums 1, 3, 6, 10 15, 21,

and, after the /?th term, we get n(n + l ) /2, which becomes very large as n increases

However, if we start to add the terms of the series

~h + + -|- ~bf- • • • -f-

we get i, l s, ¡¿, 3“i, 1 - l/2 /', The table shows that as we add more and more

terms, these partial sums become closer and closer to 1 (See also Figure 1 1 in A Preview

o f Calculus, page 8.) In fact, by adding sufficiently many terms of the series we can make the partial sums as close as we like to 1 So it seems reasonable to say that the sum of this infinite series is 1 and to write

V - = — + — -f- — + , 2" 2 4 8 16 + ■ • ■+ —1

2 + • • • = l

C o m p a re w ith th e im p ro p e r in tegral

| fix) dx = lim fix) dx

T o find th is in tegral w e in te g ra te fro m 1 to t

an d th e n let t —» Fo r a s e rie s , w e sum fro m

1 to n an d th e n let n —»

We use a similar idea to determine whether or not a general series ( 1) has a sum We

These partial sums form a new sequence {5,,}, which may or may not have a limit If

lim n_* s„ = 5 exists (as a finite number), then, as in the preceding example, we call it the sum of the infinite series 2 a n.

|~2~| Definition Given a series 2 “-i a n = a x + a 2 + a 3 + • • let s„ denote its

n th partial sum:

s„ = 2 a i = + a i + ‘ * * + a "

i=1

If the sequence {s,,} is convergent and lim,,-*» s n s exists as a real number, then

the series 2 a n is called convergent and we write

Thus the sum of a series is the limit of the sequence of partial sums So when we write

an = s we mean that by adding sufficiently many terms of the series we can get as

close as we like to the number s. Notice that

y a„ = hm 2 ai

EXAMPLE 1 An important example of an infinite series is the geometric series

a + ar + ar2 + ar7, + • • • 4- ar" 1 + * * ' = X arn 1 a 7 ^ 0

«=1

Each term is obtained from the preceding one by multiplying it by the common ratio r

(We have already considered the special case where a = 2 and r = 2 on page 565.)

jf r = ] ^ then s„ — a + a + • • • -}- ¿/ = nci ^ i t 00 Since lim,,-.» sn doesn t exist, the

geometric series diverges in this case

If r 7* 1, we have

s n = a + ar + a r2 + • • * + cirn 1

W h a t do w e re ally m ean w h e n w e s a y th a t the

sum o f th e series in E x a m p le 2 is 3? O f c o urse,

w e c a n 't litera lly ad d an infinite n u m b e r of

te rm s , on e by o n e B u t, according to D e fin i­

tio n 2 , th e to ta l sum is th e lim it o f the

s e q u e n c e o f pa rtial s u m s S o , by taking the

sum o f s u ffic ie n tly m a n y te rm s , w e can g e t as

close as w e like to the n um ber 3 T h e table

s h o w s th e first ten pa rtial sum s s„ and the

gra ph in Fig u re 2 s h o w s h o w the seq u e n c e of

p a rtial sum s a p p ro a ch e s 3

Subtracting these equations, we get

sn — rs„ = a — arn

a( 1 - r n) S" t1 — r

If —1 < /* < 1, we know from (8.1.7) that r" —> 0 as n —> oo, so

Inn = lim — - = - lim rn =

Thus when | /*| < 1 the geometric series is convergent and its sum is a /{ l — r).

If r ^ _ 1 or r > 1, the sequence {r"} is divergent by (8.1.7) and so, by Equation 3, lim,,-.* sn does not exist Therefore the geometric series diverges in those cases [

We summarize the results of Example 1 as follows

[~4~[ The geometric series

If | /* | 5s 1, the geometric series is divergent

[3 EXAMPLE 2 Find the sum of the geometric series

5 - 10 , 20 _ 40 ,

S O L U T IO N The first term is a = 5 and the common ratio is /* = — Since | r | = j < 1,

the series is convergent by (4) and its sum is

Another way to identify a and r is to write out

the first few terms:

4 + y + t + • • •

EXAMPLE 3 Is the series X 22"3' " convergent or divergent?

n~ 1 SOLUTION Let’s rewrite the nth term of the series in the form ar"~':

2 22"31_" = 2 (22)n3"<"-l) = 2 = 2 4 (f) " " 1

We recognize this series as a geometric series with a = 4 and r = \ Since r > 1, the

Q EXAMPLE 4 Expressing a repeating decimal as a rational number

Write the number 2.317 = 2.3171717 as a ratio of integers

SOLUTION

17 J 1 4- i l + 2.3171717 • • = 2.3 + -JqT + 1Qs + 107 +

After the first term we have a geometric series with a =

-= 23 J 7 _ = 1147

10 + 990 ~ 495

EXAMPLE 5 A series with variable terms ., p;n(i the sum of the series X x", where lx I < 1.n n a me ™ " 0 1 1

T , ,.,ith n = 0 and so the first term is x° = 1 (With

series, we adopt the convention that x n = 1 even w en x

i X" = 1 + X + X2 + X3 + A-4 + • • •

/1 = 0

1 r = v Since I r\ = \x\ < 1, it converges and This is a geometric series with a = 1 and / A* 011 1 ' ' ' &(4 ) gives

EXAMPLE 6 A telescoping sum Show that the series 2 , f^ n + ^ IS convergent, and

find its sum.

„ * • cn mP ao back to the definition of a convergentSOLUTKj' i This is not a geometric series, so we ua

series and com pute the partial sums.

1+ • • • + -

" , ? , / ( / + ! ) 1 - 2 2 * 3 3 - 4

SECTION 8.2 SERIES 569

Notice that the terms cancel in pairs.

This is an example of a telescoping sum:

Because of all the cancellations, the sum

collapses (like a pirates collapsing

telescope) into just two terms.

Figure 3 illustrates Example 6 by show­

ing the graphs of the sequence of terms

a„ = \/[n(n + 1)] and the sequence {5,,}

of partial sums Notice that an — > 0 and

sn — > 1 See Exercises 56 and 57 for two

geometric interpretations of Example 6.

We can simplify this expression if we use the partial fraction decomposition

/(/ + 1 ) / / + 1(see Section 5.7) Thus we have

Therefore the given series is convergent and

SOLUTION For this particular series it’s convenient to consider the partial sums s2, s4, s8,

^16, ^32, • • • anc* show that they become large

The m ethod used in Exam ple 7 for showing

th a t the harm onic series diverges is due to the

French scholar N icole Oresm e (1 3 2 3 -1 3 8 2 ).

This shows that s2-

series diverges

sc as n 00 and so {.s',,} is divergent Therefore the harmonic

[Tj Theorem If the series 2 is convergent, then lim an — 0.

PROOF Let s„ = a x + a 2 + * * * + a n- Then a„ = s„ - s n-1- Since S a n is convergent, the

sequence {5,,} is convergent Let lim,,-»* s n = s. Since n — 1 —» 00 as n —> °o, we also

have lim„-_>x s„-i = Therefore

I Note 2: The converse of Theorem 6 is not true in general If lim„ _* a„ = 0, we cannot

conclude that 2 is convergent Observe that for the harmonic series 2 \/n we have

, /- in Example 7 that 2 \/n is divergent.

cc

series 2 an is divergent.

/1=1 * 7

The Test for Divergence follows from Theorem 6 because, if the series is not divergent,

then it is convergent, and so lim,,-** a n = 0.

m Theorem If 2 a„ and 2 bn are convergent series, then so are the series 2 ca„ (where c is a constant), 2 (a„ + b„), and 2 {a„ — b,,), and

(i) i ca„ = c i a„ <»> I , (ữ" + b"] = ỉ ê a " + I , b"

(ill) ^ (û/j b n) ^ Cin b n

SECTION 8.2 SERIES 571

These properties of convergent series follow from the corresponding Limit Laws for Sequences in Section 8.1 For instance, here is how part (ii) of Theorem 8 is proved:Let

it follows that the entire series n /( n 3 + 1) is convergent Similarly, if it is known that the series 1T i = n + i an converges, then the full series

1 (a) W hat is the difference between a sequence and a series?

(b) W hat is a convergent series? What is a divergent series?

2 Explain what it means to say that a„ = 5.

S 3-8 Find at least 10 partial sums of the series Graph both the

sequence of terms and the sequence of partial sums on the same

screen Does it appear that the series is convergent or divergent?

if it is convergent, find the sum If it is divergent, explain why.

(b) Determine whether S^=i a n is convergent.

10 (a) Explain the difference between

11-18 Determine whether the geometric series is convergent or

divergent If it is convergent, find its sum.

19-30 Determine whether the series is convergent or divergent

If it is convergent, find its sum.

31-34 Determine whether the series is convergent or divergent by

expressing sn as a telescoping sum (as in Example 6) If it is con­

vergent, find its sum.

(a) Do you think that x < 1 or x = 1 ?

(b) Sum a geometric series to find the value of a (c) How many decimal representations does the number 1 have?

(d) Which numbers have more than one decimal representation?

36-40 Express the number as a ratio of integers.

36 0 7 3 = 0 7 3 7 3 7 3 7 3 .

37 0 2 = 0 2 2 2 2