Multivariable calculus 4e james stewart

Fourth Edition Reviewers Jennifer Bailey, Colorado School of Mines Lewis Blake, Duke University James Cook, North Carolina State University Costel Ionita, Dixie State College Lawrence Le

Multivariable Calculus

Concepts and Contexts | 4e

Calculus and the Architecture of Curves

The cover photograph shows the

DZ Bank in Berlin, designed and

built 1995–2001 by Frank Gehry

and Associates The interior atrium

is dominated by a curvaceous

four-story stainless steel sculptural

shell that suggests a prehistoric

creature and houses a central

con-ference space

The highly complex structures

that Frank Gehry designs would be

impossible to build without the computer

The CATIA software that his

archi-tects and engineers use to produce the

computer models is based on principles of

calculus—fitting curves by matching tangent

lines, making sure the curvature isn’t too

large, and controlling parametric surfaces

“Consequently,” says Gehry, “we have a lot

of freedom I can play with shapes.”

The process starts with Gehry’s initial

sketches, which are translated into a

succes-sion of physical models (Hundreds of different

physical models were constructed during the design

of the building, first with basic wooden blocks and then

evolving into more sculptural forms.) Then an engineer

uses a digitizer to record the coordinates of a series of

points on a physical model The digitized points are fed

into a computer and the CATIA software is used to link

these points with smooth curves (It joins curves so that

their tangent lines coincide; you can use the same idea to

design the shapes of letters in the Laboratory Project on

page 208 of this book.) The architect has considerable

free-dom in creating these curves, guided by displays of the

curve, its derivative, and its curvature Then the curves are

connected to each other by a parametric surface,

and again the architect can do so in many possible

ways with the guidance of displays of the geometric

characteristics of the surface

The CATIA model is then used to produce

another physical model, which, in turn, suggests

modifications and leads to additional computer

and physical models

The CATIA program was developed in France

by Dassault Systèmes, originally for designing airplanes, and was subsequently employed in the automotive industry Frank Gehry, because of his complex sculptural shapes, is the first to use

it in architecture It helps him answer his tion, “How wiggly can you get and still make a building?”

Multivariable Calculus

Concepts and Contexts | 4e

Australia Brazil Japan Korea Mexico Singapore Spain United Kingdom United States

James Stewart

McMaster University

and University of Toronto

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8.3 The Integral and Comparison Tests; Estimating Sums 575

8.4 Other Convergence Tests 585

8.5 Power Series 592

8.6 Representations of Functions as Power Series 598

8.7 Taylor and Maclaurin Series 604

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

8.8 Applications of Taylor Polynomials 619

Applied Project ■ Radiation from the Stars 627

Review 628

Focus on Problem Solving 631

9.1 Three-Dimensional Coordinate Systems 634

9.2 Vectors 639

9.3 The Dot Product 648

9.4 The Cross Product 654

Discovery Project ■ The Geometry of a Tetrahedron 662

9.5 Equations of Lines and Planes 663

Laboratory Project ■ Putting 3D in Perspective 672

9.6 Functions and Surfaces 673

9.7 Cylindrical and Spherical Coordinates 682

Laboratory Project ■ Families of Surfaces 687

Review 688

Focus on Problem Solving 691

9 Vectors and the Geometry of Space 633

8 Infinite Sequences and Series 553

Preface xi

To the Student xx

10.1 Vector Functions and Space Curves 694

10.2 Derivatives and Integrals of Vector Functions 701

10.3 Arc Length and Curvature 707

10.4 Motion in Space: Velocity and Acceleration 716

Applied Project ■ Kepler’s Laws 726

10.5 Parametric Surfaces 727Review 733

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

11.5 The Chain Rule 780

11.6 Directional Derivatives and the Gradient Vector 789

11.7 Maximum and Minimum Values 802

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

11.8 Lagrange Multipliers 813

Applied Project ■ Rocket Science 820 Applied Project ■ Hydro-Turbine Optimization 821

Review 822

Focus on Problem Solving 827

12.1 Double Integrals over Rectangles 830

12.2 Iterated Integrals 838

12.3 Double Integrals over General Regions 844

12.4 Double Integrals in Polar Coordinates 853

12.5 Applications of Double Integrals 858

12.6 Surface Area 868

12.7 Triple Integrals 873

Discovery Project ■ Volumes of Hyperspheres 883

12.8 Triple Integrals in Cylindrical and Spherical Coordinates 883

Applied Project ■ Roller Derby 889 Discovery Project ■ The Intersection of Three Cylinders 890

12 Multiple Integrals 829

11 Partial Derivatives 737

10 Vector Functions 693

Writing Project ■ Three Men and Two Theorems 966

13.8 The Divergence Theorem 967

13.9 Summary 973Review 974

Focus on Problem Solving 977

D Precise Definitions of Limits A2

E A Few Proofs A3

H Polar Coordinates A6

I Complex Numbers A22

J Answers to Odd-Numbered Exercises A31

Index A51 Appendixes A1

13 Vector Calculus 905

When the first edition of this book appeared twelve years ago, a heated debate about culus 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 mathematicsdepartments Since then the rhetoric has calmed down somewhat as reformers and tradi-tionalists have realized that they have a common goal: to enable students to understand andappreciate calculus.

cal-The first three editions were intended to be a synthesis of reform and traditionalapproaches to calculus instruction In this fourth edition I continue to follow that path byemphasizing conceptual understanding through visual, verbal, numerical, and algebraicapproaches I aim to convey to the student both the practical power of calculus and theintrinsic 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 exampleshave 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 I 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 porate binomial series into 8.7

incor-■ More than 25% of the exercises in each chapter are new Here are a few of myfavorites: 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 onpage 736, and Problem 11 on page 904

Features Conceptual Exercises 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 beginwith requests to explain the meanings of the basic concepts of the section (See, forinstance, the first couple of exercises in Sections 8.2, 11.2, and 11.3 I often use them as

Preface

xi

a basis for classroom discussions.) Similarly, review sections begin with a Concept Checkand a True-False Quiz Other exercises test conceptual understanding through graphs ortables (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)

Graded Exercise Sets Each exercise set is carefully graded, progressing from basic conceptual exercises and

skill-development problems to more challenging problems involving applications andproofs

Real-World Data My assistants and I have spent a great deal of time looking in libraries, contacting

compa-nies 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 theexamples 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 tives are introduced in Section 11.3 by examining a column in a table of values of the heatindex (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 byusing a temperature contour map to estimate the rate of change of temperature at Reno

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

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

Projects One way of involving students and making them active learners is to have them work

(per-haps in groups) on extended projects that give a feeling of substantial accomplishment

when completed Applied Projects involve applications that are designed to appeal to the

imagination of students The project after Section 11.8 uses Lagrange multipliers to mine the masses of the three stages of a rocket so as to minimize the total mass while

deter-enabling the rocket to reach a desired velocity Discovery Projects explore aspects of

geometry: tetrahedra (after Section 9.4), hyperspheres (after Section 12.7), and

intersec-tions of three cylinders (after Section 12.8) The Laboratory Project on page 687 uses

tech-nology 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

inter-actions of the three men involved Many additional projects are provided in the tor’s Guide.

Instruc-Technology The availability of technology makes it not less important but more important to

under-stand clearly the concepts that underlie the images on the screen But, when properly used,graphing calculators and computers are powerful tools for discovering and understandingthose concepts I assume that the student has access to either a graphing calculator or acomputer algebra system The icon ;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 otherexercises as well The symbol is reserved for problems in which the full resources of acomputer algebra system (like Derive, Maple, Mathematica, or the TI-89/92) are required.But technology doesn’t make pencil and paper obsolete Hand calculation and sketches areoften preferable to technology for illustrating and reinforcing some concepts Both instruc-tors and students need to develop the ability to decide where the hand or the machine isappropriate

CAS

PREFACE xiii

Tools for Enriching™ Calculus TEC is a companion to the text and is intended to enrich and complement its contents (It

is now accessible from the Internet at www.stewartcalculus.com.) Developed by Harvey

Keynes, Dan Clegg, Hubert Hohn, and myself, TEC uses a discovery and exploratoryapproach In sections of the book where technology is particularly appropriate, marginalicons direct students to TEC modules that provide a laboratory environment in which theycan explore the topic in different ways and at different levels Visuals are animations of fig-ures in the text; Modules are more elaborate activities and include exercises Instructorscan choose to become involved at several different levels, ranging from simply encourag-ing students to use the Visuals and Modules for independent exploration, to assigning spe-cific exercises from those included with each Module, or to creating additional exercises,labs, and projects that make use of the Visuals and Modules

TEC also includes Homework Hints for representative exercises (usually

odd-num-bered) in every section of the text, indicated by printing the exercise number in red Thesehints are usually presented in the form of questions and try to imitate an effective teachingassistant 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

Enhanced WebAssign Technology is having an impact on the way homework is assigned to students,

particu-larly in large classes The use of online homework is growing and its appeal depends onease of use, grading precision, and reliability With the fourth edition we have been work-ing with the calculus community and WebAssign to develop an online homework system.Many of the exercises in each section are assignable as online homework, including freeresponse, multiple choice, and multi-part formats The system also includes Active Examples,

in which students are guided in step-by-step tutorials through text examples, with links tothe textbook and to video solutions

Website: www.stewartcalculus.com 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 inTaylor Series, Linear Differential Equations, Second-Order Linear DifferentialEquations, 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 prioreditions)

Content

8■Infinite Sequences and Series

Tests for the convergence of series are considered briefly, with intuitive rather than mal justifications Numerical estimates of sums of series are based on which test wasused to prove convergence The emphasis is on Taylor series and polynomials and theirapplications to physics Error estimates include those from graphing devices

for-9■Vectors and The Geometry of Space

The dot product and cross product of vectors are given geometric definitions, motivated bywork 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 para-metric curves in Chapter 1, parametric surfaces are introduced as soon as possible, namely,

in this chapter I think an early familiarity with such surfaces is desirable, especially withthe capability of computers to produce their graphs Then tangent planes and areas of para-metric 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 braic points of view In particular, I introduce partial derivatives by looking at a specificcolumn in a table of values of the heat index (perceived air temperature) as a function ofthe actual temperature and the relative humidity Directional derivatives are estimated fromcontour maps of temperature, pressure, and snowfall

alge-12■Multiple Integrals

Contour maps and the Midpoint Rule are used to estimate the average snowfall and averagetemperature in given regions Double and triple integrals are used to compute probabil-ities, 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 Baywind patterns The similarities among the Fundamental Theorem for line integrals, Green’sTheorem, Stokes’ Theorem, and the Divergence Theorem are emphasized

Ancillaries

Multivariable Calculus: Concepts and Contexts, Fourth Edition, is supported by a

com-plete set of ancillaries developed under my direction Each piece has been designed toenhance student understanding and to facilitate creative instruction The table on pagesxviii 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 Edition Reviewers

Jennifer Bailey, Colorado School of Mines

Lewis Blake, Duke University

James Cook, North Carolina State University

Costel Ionita, Dixie State College

Lawrence Levine, Stevens Institute of Technology

Scott Mortensen, Dixie State College

Drew Pasteur, North Carolina State University Jeffrey Powell, Samford University

Barbara Tozzi, Brookdale Community College Kathryn Turner, Utah State University Cathy Zucco-Tevelof, Arcadia University

PREFACE xv

Previous Edition Reviewers

Irfan Altas, Charles Sturt University

William Ardis, Collin County Community College

Barbara Bath, Colorado School of Mines

Neil Berger, University of 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 Broadwin, Jericho High School (retired)

Charles Bu, Wellesley University

Meghan Anne Burke, Kennesaw State University

Robert Burton, Oregon State University

Roxanne M Byrne, University of Colorado at Denver

Maria E Calzada, Loyola University – New Orleans

Larry Cannon, Utah State University

Deborah Troutman Cantrell,

Chattanooga State Technical Community College

Bem Cayco, San Jose State University

John Chadam, University of Pittsburgh

Robert A Chaffer, Central Michigan University

Dan Clegg, Palomar College

Camille P Cochrane, Shelton State Community College

James Daly, University of Colorado

Richard Davis, Edmonds Community College

Susan Dean, DeAnza College

Richard DiDio, LaSalle University

Robert Dieffenbach, Miami University – Middletown

Fred Dodd, University of South Alabama

Helmut Doll, Bloomsburg University

William Dunham, Muhlenberg College

David A Edwards, The University of 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 of 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 of Georgia

Daniel Grayson,

University of Illinois at Urbana–Champaign

Raymond Greenwell, Hofstra University Gerrald Gustave Greivel, Colorado School of 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 of South Alabama

Judy Holdener, United States Air Force Academy Randall R Holmes, Auburn University

Barry D Hughes, University of Melbourne Mike Hurley, Case Western Reserve University Gary Steven Itzkowitz, Rowan University Helmer Junghans, Montgomery College Victor Kaftal, University of Cincinnati Steve Kahn, Anne Arundel Community College

Mohammad A Kazemi,

University of North Carolina, Charlotte

Harvey Keynes, University of Minnesota Kandace Alyson Kling, Portland Community College Ronald Knill, Tulane University

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

William L Lepowsky, Laney College Kathryn Lesh, University of Toledo Estela Llinas, University of Pittsburgh at Greensburg

Beth Turner Long,

Pellissippi State Technical Community College

Miroslav Lovri´c, 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 of New Brunswick

Jim McKinney,

California State Polytechnic University–Pomona

Richard Eugene Mercer, Wright State University

David Minda, University of 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 of Kansas

David Patocka, Tulsa Community College–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

Hernan Rivera, Texas Lutheran University

Richard Rochberg, Washington University

Gil Rodriguez, Los Medanos College

David C Royster, University of 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 Bernd S.W Schroeder, Louisiana Tech University Jeffrey Scott Scroggs, North Carolina State University James F Selgrade, North Carolina State University Brad Shelton, University of Oregon

Don Small,

United States Military Academy –West Point

Linda E Sundbye, The Metropolitan State College of Denver Richard B Thompson,The University of Arizona

William K Tomhave, Concordia College Lorenzo Traldi, Lafayette College

Alan Tucker,

State University of New York at Stony Brook

Tom Tucker, Colgate University George Van Zwalenberg, Calvin College Dennis Watson, Clark College

Paul R Wenston, The University of Georgia Ruth Williams, University of California–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 of 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 GailWolkowicz for their advice and suggestions; Al Shenk and Dennis Zill for permission touse 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, andPaul Sally for ideas for exercises; Dan Drucker for the roller derby project; and TomFarmer, Fred Gass, John Ramsay, Larry Riddle, V K Srinivasan, and Philip Straffin forideas for projects I’m grateful to Dan Clegg, Jeff Cole, and Tim Flaherty for preparing theanswer manuscript and suggesting ways to improve the exercises

As well, I thank those who have contributed to past editions: Ed Barbeau, GeorgeBergman, David Bleecker, Fred Brauer, Andy Bulman-Fleming, Tom DiCiccio, MartinErickson, 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 Rosen-thal, Carl Riehm, Ira Rosenholtz, Doug Shaw, Dan Silver, Lowell Smylie, Larry Wallen,Saleem Watson, and Alan Weinstein

PREFACE xvii

I also thank Stephanie Kuhns, Rebekah Million, Brian Betsill, and Kathi Townes ofTECH-arts for their production services; Marv Riedesel and Mary Johnson for their care-ful 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; JayCampbell, senior developmental editor; Jeannine Lawless, associate editor; ElizabethNeustaetter, editorial assistant; Bob Kauser, permissions editor; Becky Cross, print / mediabuyer; Vernon Boes, art director; Rob Hugel, creative director; and Irene Morris, coverdesigner They have all done an outstanding job

I have been very fortunate to have worked with some of the best mathematics editors in thebusiness 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

JAMES STEWART

xviii |||| Electronic items |||| Printed items

PowerLecture CD-ROM with JoinIn and ExamView

ISBN 0-495-56049-9

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 of

the Instructor’s Guide Also contains JoinIn on TurningPoint

personal response system questions and ExamView

algor-ithmic test generation See below for complete descriptions.

Tools for Enriching™ Calculus

by James Stewart, Harvey Keynes, Dan Clegg,

and developer Hu Hohn

TEC provides a laboratory environment in which students

can explore selected topics TEC also includes homework

hints for representative exercises Available online at

www.stewartcalculus.com

Instructor’s Guide

by Douglas Shaw and James Stewart

ISBN 0-495-56047-2

Each section of the main text is discussed from several

view-points and contains suggested time to allot, view-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

ISBN 0-495-56059-6

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

ISBN 0-495-56056-1

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

Printed Test Bank

by William Tomhave and Xuequi Zeng

ISBN 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 of problems and complete questions from the Printed Test Bank.

JoinIn on TurningPoint

Enhance how your students interact with you, your lecture, and each other Brooks /Cole, Cengage Learning is now pleased to offer you book-specific content for Response Systems tailored

to Stewart’s Calculus, allowing you to transform your room and assess your students’ progress with instant in-class quizzes and polls Contact your local Cengage representative

class-to learn more about JoinIn on TurningPoint and our exclusive infrared and radio-frequency hardware solutions.

Text-Specific DVDs

ISBN 0-495-56050-2

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.

Solution Builder

www.cengage.com/solutionbuilder

The online Solution Builder lets instructors easily build and save personal solution sets either for printing or posting on password-protected class websites Contact your local sales representative for more information on obtaining an account for this instructor-only resource.

eBook Option

ISBN 0-495-56121-5

Whether you prefer a basic downloadable eBook or a mium multimedia eBook with search, highlighting, and note taking capabilities as well as links to videos and simulations, this new edition offers a range of eBook options to fit how you want to read and interact with the content.

pre-Stewart Specialty Website

www.stewartcalculus.com

Contents: Algebra Review N Additional Topics N Drill exercises N Challenge Problems N Web Links N History of Mathematics N Tools for Enriching Calculus (TEC)

Ancillaries for Instructors and Students TEC

Ancillaries for Instructors

Enhanced WebAssign

Instant feedback, grading precision, and ease of use are just

three reasons why WebAssign is the most widely used

home-work system in higher education WebAssign’s homehome-work

delivery system lets instructors deliver, collect, grade and

record assignments via the web And now, this proven system

has been enhanced to include end-of-section problems from

Stewart’s Calculus: Concepts and Contexts—incorporating

exercises, examples, video skillbuilders and quizzes to promote

active learning and provide the immediate, relevant feedback

students want.

The Brooks/Cole Mathematics Resource Center Website

www.cengage.com/math

When you adopt a Brooks/Cole, Cengage Learning

mathe-matics text, you and your students will have access to a

var-iety 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.

Maple CD-ROM

ISBN 0-495-01492-3 (Maple 10)

ISBN 0-495-39052-6 (Maple 11)

Maple provides an advanced, high performance

mathema-tical computation engine with fully integrated numerics &

symbolics, all accessible from a WYSIWYG technical

docu-ment environdocu-ment Available for bundling with your Stewart

Calculus text at a special discount.

Tools for Enriching™ Calculus

by James Stewart, Harvey Keynes, Dan Clegg,

and developer Hu Hohn

TEC provides a laboratory environment in which students

can explore selected topics TEC also includes homework

hints for representative exercises Available online at

www.stewartcalculus.com.

Study Guide, Multivariable

by Robert Burton and Dennis Garity

ISBN 0-495-56057-X

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

the ideas of the section, and worked-out examples with tips

on how to find the solution.

Student Solutions Manual, Multivariable

by Dan Clegg ISBN 0-495-56055-3

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 ISBN 0-495-56058-8

CalcLabs with Mathematica, Multivariable

by Selwyn Hollis ISBN 0-495-82722-3

Each of 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

ISBN 0-495-01124-X

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 for calculus courses that integrate the review of pre- calculus concepts or for 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 ISBN 0-534-25248-6

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

TEC

Student Resources

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 and you

obtain , then you’re right and rationalizing the

denominator will show that the answers are equivalent

The icon;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

1兾(1⫹ s2)

s2⫺ 1

encounter.) But that doesn’t mean that graphing devices can’t

be used to check your work on the other exercises as well Thesymbol is reserved for problems in which the full resources

of a computer algebra system (like Derive, Maple, matica, or the TI-89/92) are required

Mathe-You will also encounter the symbol |, which warns youagainst committing an error I have placed this symbol in themargin in situations where I have observed that a large propor-tion of my students tend to make the same mistake

Tools for Enriching Calculus, which is a companion to this

text, is referred to by means of the symbol and can beaccessed from www.stewartcalculus.com It directs you to mod-ules in which you can explore aspects of calculus for which the

computer is particularly useful TEC also provides Homework

Hints for representative exercises that are indicated by printing

the exercise number in red: 15.These homework hints ask youquestions that allow you to make progress toward a solutionwithout actually giving you the answer You need to pursueeach hint in an active manner with pencil and paper to work out the details If a particular hint doesn’t enable you to solvethe problem, you can click to reveal the next hint

I recommend that you keep this book for reference purposesafter 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 subsequentcourses And, because this book contains more material thancan be covered in any one course, it can also serve as a valuableresource 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 youwill discover that it is not only useful but also intrinsicallybeautiful

JA M E S S T E WA RT

TEC

CAS

Infinite Sequences and Series

Infinite sequences and series were introduced briefly in A 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 (Recall that we have previously been unable to do this.) Many of the

functions that arise in mathematical physics and chemistry, such as Bessel

func-tions, 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.

e ⫺x2

8

A sequence can be thought of as a list of numbers written in a definite order:

The number is called the first term, is the second term, and in general is the nth term We will deal exclusively with infinite sequences and so each term will have a successor

Notice that for every positive integer there is a corresponding number and so asequence can be defined as a function whose domain is the set of positive integers But weusually write instead of the function notation for the value of the function at thenumber

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

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 doesn’t have to start at 1

(a)

(b)

(c)

(d)

Find a formula for the general term 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; ingeneral, the th term will have numerator n n⫹ 2 The denominators are the powers of 5,

a5苷 73125

a4苷 ⫺ 6625

a3苷 5125

a2苷 ⫺ 4

25

a1苷 35

SECTION 8.1 SEQUENCES 555

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

so we need to multiply by a power of In Example 1(b) the factor meant westarted with a negative term Here we want to start with a positive term and so we use

or Therefore

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

(a) The sequence , where is the population of the world as of January 1 in theyear

(b) If we let be the digit in the nth decimal place of the number , then is a defined sequence whose first few terms are

well-(c) The Fibonacci sequence is defined recursively by the conditions

Each term is the sum of the two preceding terms The first few terms are

This sequence arose when the 13th-century Italian mathematician known as Fibonaccisolved a problem concerning the breeding of rabbits (see Exercise 47)

A sequence such as the one in Example 1(a), , can be pictured either byplotting its terms on a number line, as in Figure 1, or by plotting its graph, as in Figure 2.Note that, since a sequence is a function whose domain is the set of positive integers, itsgraph consists of isolated points with coordinates

From Figure 1 or Figure 2 it appears that the terms of the sequence areapproaching 1 as becomes large In fact, the difference

can be made as small as we like by taking sufficiently large We indicate this by writing

In general, the notation

means that the terms of the sequence approach as becomes large Notice that thefollowing 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

n L

a¶=

Definition A sequence has the limit and we write

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

If 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 thelimit

If you compare Definition 1 with Definition 2.5.4 you will see that the only differencebetween and is that is required to be an integer Thus

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

In particular, since we know from Section 2.5 that when wehave

if

If becomes large as n becomes large, we use the notation

In this case the sequence is divergent, but in a special way We say that diverges

to The Limit Laws given in Section 2.3 also hold for the limits of sequences and theirproofs are similar

y

1 3 4 L

a n

a nlL as n l⬁or

A more precise definition of the limit of a

sequence is given in Appendix D.

SECTION 8.1 SEQUENCES 557

If and are convergent sequences and is a constant, then

The Squeeze Theorem can also be adapted for sequences as follows (see Figure 5)

Another useful fact about limits of sequences is given by the following theorem, whichfollows from the Squeeze Theorem because

Find

and denominator by the highest power of that occurs in the denominator and then usethe Limit Laws

Here we used Equation 3 with

Applying l’Hospital’s Rule to a related function Calculate

can’t apply l’Hospital’s Rule directly because it applies not to sequences but to functions

n l⬁

lim

n l⬁

ln n n

Limit Laws for Sequences

Squeeze Theorem for Sequences

FIGURE 5

The sequence b is squeezed

between the sequences a

This shows that the guess we made earlier

from Figures 1 and 2 was correct.

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

and obtain

Therefore, by Theorem 2, we have

Determine whether the sequence is convergent or divergent

The graph of this sequence is shown in Figure 6 Since the terms oscillate between 1 andinfinitely often, does not approach any number Thus does not exist;that is, the sequence is divergent

Using the Squeeze Theorem Discuss the convergence of the sequence

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

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

The graph of the sequence in Example 7 is

shown in Figure 7 and supports the answer.

We know that as Therefore as by the Squeeze Theorem.

Limit of a geometric sequence For what values of is the sequence convergent?

putting and using Theorem 2, we have

For the cases and we have

and

and therefore by Theorem 4 If , then diverges as inExample 6 Figure 9 shows the graphs for various values of (The case isshown in Figure 6.)

n

an

1 1

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

and graphing in dot mode, starting with

and setting the -step equal to The result is

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

The sequence is convergent if and divergent for all othervalues of

It is called decreasing if for all A sequence

is monotonic if it is either increasing or decreasing.

The sequence is decreasing because

and so for all

Show that the sequence is decreasing

This inequality is equivalent to the one we get by cross-multiplication:

Since , we know that the inequality is true Therefore and

so is decreasing

Thus is decreasing on f 1, ⬁and so f n ⬎ f n ⫹ 1 Therefore a nis decreasing

SECTION 8.1 SEQUENCES 561

It is bounded below if there is a number such that

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

For instance, the sequence is bounded below but not above Thesequence is bounded because for all

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

satisfies but is divergent, from Example 6] and not every tonic sequence is convergent But if a sequence is both bounded and

mono-monotonic, then it must be convergent This fact is stated without proof as Theorem 8, butintuitively you can understand why it is true by looking at Figure 10 If is increasingand for all , then the terms are forced to crowd together and approach some num-ber

convergent

The limit of a recursively defined sequence Investigate the sequence

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 showthat for all This is true for because If we assumethat it is true for , then we have

Mathematical induction is often used in

deal-ing with recursive sequences See page 84 for

a discussion of the Principle of Mathematical

Induction.

We have deduced that is true for Therefore the inequality is truefor all by induction.

Next we verify that is bounded by showing that for all (Since thesequence is increasing, we already know that it has a lower bound: for all ) We know that , so the assertion is true for Suppose it is true for Then

so

andThus

This shows, by mathematical induction, that for all Since the sequence is increasing and bounded, the Monotonic Sequence Theoremguarantees that it has a limit The theorem doesn’t tell us what the value of the limit is.But now that we know exists, we can use the given recurrence relation towrite

Since , it follows that too (as , also) So we have

Solving this equation for , we get L L苷 6, as we predicted

1. (a) What is a sequence?

(b) What does it mean to say that ?

(c) What does it mean to say that ?

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

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

4. List the first nine terms of the sequence 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 of the sequence,

assuming that the pattern of the first few terms continues.

11–34 Determine whether the sequence converges or diverges

If it converges, find the limit.

;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 years the investment is worth

dollars.

(a) Find the first five terms of the sequence

(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 months is given by

the sequence

(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 after months is

e n其 44. Find the first 40 terms of the sequence defined by

and Do the same if 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 ?

46. (a) If , what is the value of ? (b) A sequence 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.

(c) 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).

47. (a) 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 month? Show that the answer is , where is the Fibonacci sequence defined in Example 3(c).

Assuming that is convergent, find its limit.

48. Find the limit of the sequence

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

or not monotonic Is the sequence bounded?

53. Suppose you know that 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?

54. A sequence is given by , (a) By induction or otherwise, show that is increasing and bounded above by 3 Apply the Monotonic Sequence Theorem to show that exists.

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 terms of this sequence starting with an initial tion Use this program to do the following.

popula-1. Calculate 20 or 30 terms of the sequence for and for two values of such that

Graph each sequence Do the sequences appear to converge? Repeat for a ferent value of between 0 and 1 Does the limit depend on the choice of ? Does it depend on the choice of ?

dif-2. Calculate terms of the sequence for a value of between 3 and 3.4 and plot them What do you notice about the behavior of the terms?

k k

55. Show that the sequence defined by

is increasing and for all Deduce that is

conver-gent and find its limit.

56. Show that the sequence defined by

satisfies and is decreasing Deduce that the

sequence is convergent and find its limit.

57. We know that [from (7) with ]

Use logarithms to determine how large has to be so

that

, where is a continuous function If , show that

(b) Illustrate part (a) by taking , , and

estimating the value of to five decimal places.

59. The size of an undisturbed fish population has been modeled

environ-(b) Show that (c) Use part (b) to show that if , then ;

in other words, the population dies out.

(d) Now assume that Show that if , then

is increasing and Show also that

if , then is decreasing and Deduce that if , then

60. A sequence is defined recursively by

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

This gives the continued fraction expansion

expres-which is called an infinite series (or just a series) and is denoted, for short, by the symbol

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

because if we start adding the terms we get the cumulative sums 1, 3, 6, 10, 15, 21, and, after the term, we get , which becomes very large as increases.However, if we start to add the terms of the series

we get , , , , , , , , The table shows that as we add more and more

terms, these partial sums become closer and closer to 1 (See also Figure 11 in A Preview

of 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 thisinfinite series is 1 and to write

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

4. For values of 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 by 0.001? This type of behavior is

called chaotic and is exhibited by insect populations under certain conditions.

The current record is that has been computed

to (more than a trillion)

decimal places by Shigeru Kondo and his

collaborators.

1,241,100,000,000

␲

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

consider the partial sums

and, in general,

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

exists (as a finite number), then, as in the preceding example, we call it thesum of the infinite series

th partial sum:

If the sequence is convergent and exists as a real number, thenthe series is called convergent and we write

The number is called the sum of the series If the sequence is divergent, then the

series is called divergent.

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

we mean that by adding sufficiently many terms of the series we can get asclose as we like to the number Notice that

An important example of an infinite series is the geometric series

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

(We have already considered the special case where and on page 565.)

geometric series diverges in this case

a苷1 2

Compare with the improper integral

To find this integral we integrate from 1 to

and then let For a series, we sum from

1 to and then let n n l

SECTION 8.2 SERIES 567

Subtracting these equations, we get

If , we know from (8.1.7) that as , so

Thus when the geometric series is convergent and its sum is

If or , the sequence is divergent by (8.1.7) and so, by Equation 3,does not exist Therefore the geometric series diverges in those cases

We summarize the results of Example 1 as follows

The geometric series

is convergent if and its sum is

If , the geometric series is divergent

Find the sum of the geometric series

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

FIGURE 2

sn

20 3

Figure 1 provides a geometric demonstration

of the result in Example 1 If the triangles are

constructed as shown and is the sum of the

series, then, by similar triangles,

a

ar a-ar

What do we really mean when we say that the

sum of the series in Example 2 is ? Of course,

we can’t literally add an infinite number of

terms, one by one But, according to

Defini-tion 2, the total sum is the limit of the

sequence of partial sums So, by taking the

sum of sufficiently many terms, we can get as

close as we like to the number The table

shows the first ten partial sums and the

graph in Figure 2 shows how the sequence of

partial sums approaches 3

s n

3 3

Is the series convergent or divergent?

We recognize this series as a geometric series with and Since , theseries diverges by (4)

Expressing a repeating decimal as a rational number

Write the number as a ratio of integers

SOLUTION

After the first term we have a geometric series with and Therefore

A series with variable terms Find the sum of the series , where

series, we adopt the convention that even when ) Thus

This is a geometric series with and Since , it converges and(4) gives

A telescoping sum Show that the series is convergent, and find its sum

series and compute the partial sums.

Another way to identify and is to write out

the first few terms:

4  16

3  64

9    

r a

Module 8.2 explores a series that

depends on an angle in a triangle and enables

you to see how rapidly the series converges

when varies

TEC

SECTION 8.2 SERIES 569

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

(see Section 5.7) Thus we have

and so

Therefore the given series is convergent and

Show that the harmonic series

is divergent

, and show that they become large

Similarly, , , and in general

This shows that as and so is divergent Therefore the harmonicseries diverges

s32

5 2

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 pirate’s collapsing

telescope) into just two terms.

Figure 3 illustrates Example 6 by

show-ing the graphs of the sequence of terms

and the sequence

of partial sums Notice that and

See Exercises 56 and 57 for two

geometric interpretations of Example 6.

The method used in Example 7 for showing

that the harmonic series diverges is due to the

French scholar Nicole Oresme (1323–1382).

Theorem If the series is convergent, then

sequence is convergent Let Since as , we also

Note 1: With any series we associate two sequences: the sequence of its tial sums and the sequence of its terms If is convergent, then the limit of thesequence is (the sum of the series) and, as Theorem 6 asserts, the limit of thesequence is 0

par-| Note 2: The converse of Theorem 6 is not true in general If , we cannotconclude that is convergent Observe that for the harmonic series we have

as , but we showed in Example 7 that is divergent

So the series diverges by the Test for Divergence

Note 3: If we find that , we know that is divergent If we find that

, we know nothing about the convergence or divergence of Rememberthe warning in Note 2: If , the series might converge or it mightdiverge

(where is a constant), , and , and

SECTION 8.2 SERIES 571

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

The nth partial sum for the series is

and, using Equation 5.2.10, we have

Therefore is convergent and its sum is

Find the sum of the series

In Example 6 we found that

So, by Theorem 8, the given series is convergent and

Note 4: A finite number of terms doesn’t affect the convergence or divergence of aseries For instance, suppose that we were able to show that the series

11 2

苷 1

r苷1 2

a苷1 2