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Alternating Series Test Practice Problems: Mastering Convergence and Divergence
Author: Dr. Evelyn Reed, PhD in Mathematics, Professor of Calculus at the University of California, Berkeley. Dr. Reed has over 20 years of experience teaching calculus and has published numerous articles on advanced calculus topics.
Keyword: alternating series test practice problems
Abstract: This article provides a comprehensive exploration of alternating series test practice problems, highlighting common challenges students face and offering strategies to overcome them. We delve into the theoretical underpinnings of the Alternating Series Test, illustrate its application through diverse examples, and discuss advanced problem-solving techniques. The article also addresses misconceptions and provides ample practice problems with detailed solutions.
Introduction: The Alternating Series Test is a crucial tool in the calculus arsenal for determining the convergence or divergence of infinite series. Understanding and effectively applying this test is vital for students pursuing advanced mathematics, engineering, physics, and other STEM fields. However, mastering the alternating series test practice problems requires more than just memorizing the theorem; it necessitates a deep understanding of its conditions and nuances. This article aims to bridge the gap between theoretical knowledge and practical application, offering a structured approach to tackling a wide range of alternating series test practice problems.
Understanding the Alternating Series Test:
The Alternating Series Test states that an alternating series of the form ∑ (-1)^(n+1) b_n (where b_n > 0 for all n) converges if:
1. b_(n+1) ≤ b_n for all n (the terms are non-increasing)
2. lim (n→∞) b_n = 0 (the terms approach zero)
Failing to satisfy either condition implies the series may diverge. It’s crucial to understand that the test only provides sufficient conditions for convergence; a series may converge even if it doesn't satisfy these conditions.
Common Challenges in Alternating Series Test Practice Problems:
Many students struggle with alternating series test practice problems due to several common challenges:
1. Identifying Alternating Series: The first hurdle is recognizing when a series is indeed an alternating series. Students need to be comfortable manipulating series expressions to identify the alternating component, often involving (-1)^n or (-1)^(n+1) terms.
2. Determining Monotonicity (b_(n+1) ≤ b_n): Proving that the terms are non-increasing can be challenging. This often requires techniques from precalculus and calculus, including derivatives and inequalities. Students may struggle to find suitable methods to show monotonicity or may make mistakes in their algebraic manipulations.
3. Evaluating Limits (lim (n→∞) b_n = 0): Correctly evaluating the limit of the terms as n approaches infinity is essential. This step frequently involves using L'Hôpital's rule, limit theorems, or other limit evaluation techniques. Errors in limit calculations can lead to incorrect conclusions.
4. Distinguishing between the Alternating Series Test and Other Convergence Tests: Students often confuse the Alternating Series Test with other convergence tests, such as the Ratio Test, Root Test, or Integral Test. Understanding the strengths and limitations of each test is paramount for choosing the appropriate method for a given problem.
5. Interpreting Results: Even after correctly applying the test, students may misinterpret the results. A conclusive statement indicating convergence or divergence is crucial.
Strategies for Success with Alternating Series Test Practice Problems:
To overcome these challenges, students should:
1. Practice, Practice, Practice: The most effective way to master alternating series test practice problems is through consistent practice. Work through numerous examples, starting with simple problems and gradually increasing the complexity.
2. Develop a Systematic Approach: Create a step-by-step procedure to tackle these problems. This should include clearly identifying the terms b_n, checking for monotonicity, evaluating the limit, and drawing a definitive conclusion.
3. Utilize Visual Aids: Graphs can be incredibly helpful in understanding the behavior of the terms b_n and visualizing whether they are non-increasing.
4. Seek Help When Needed: Don't hesitate to ask for assistance from teachers, tutors, or classmates when encountering difficulties. Explaining the problem aloud can often help identify areas of confusion.
5. Review Precalculus and Calculus Fundamentals: Strong foundational knowledge of precalculus and calculus concepts is crucial for success. Review limits, inequalities, and differentiation techniques as needed.
Advanced Alternating Series Test Practice Problems:
Some problems might involve more complex scenarios requiring additional techniques:
1. Series involving multiple terms: Series that are a combination of alternating and non-alternating components might need manipulation before applying the alternating series test.
2. Conditional vs. Absolute Convergence: Students need to understand the distinction between conditional and absolute convergence. The alternating series test only establishes conditional convergence; determining absolute convergence requires applying other tests.
3. Error Estimation: The alternating series estimation theorem allows for estimation of the remainder when approximating the sum of a convergent alternating series. Practice problems involving error estimations are valuable.
Conclusion:
Mastering alternating series test practice problems requires a combination of theoretical understanding, practical application, and persistent effort. By addressing the common challenges and employing the suggested strategies, students can develop the skills necessary to confidently tackle a wide array of problems and gain a deeper appreciation for the power and elegance of the Alternating Series Test. Consistent practice and a systematic approach are key to success in this area of calculus.
FAQs:
1. What if the terms are not strictly decreasing, but only non-increasing? The Alternating Series Test still applies; the condition is non-increasing (b_(n+1) ≤ b_n), not strictly decreasing.
2. Can the Alternating Series Test be used for non-alternating series? No, the Alternating Series Test is specifically designed for alternating series. Other convergence tests should be considered for non-alternating series.
3. What if lim (n→∞) b_n ≠ 0? If the limit of the terms does not approach zero, the series diverges by the nth term test for divergence.
4. How do I determine if a series is absolutely convergent? Use tests like the comparison test, limit comparison test, ratio test, or root test to determine if the absolute value of the series converges.
5. What is the alternating series estimation theorem? It states that the error in approximating the sum of a convergent alternating series by its nth partial sum is less than or equal to the absolute value of the (n+1)th term.
6. Can I use L'Hôpital's rule to evaluate lim (n→∞) b_n? Yes, if b_n is in a form suitable for L'Hôpital's rule (typically an indeterminate form like 0/0 or ∞/∞). Remember that L'Hôpital's rule applies to functions, so you might need to consider the continuous counterpart of the sequence.
7. How do I handle series with factorial terms? Often, the ratio test is more efficient than the alternating series test for series with factorial terms.
8. What are some common mistakes students make when applying the Alternating Series Test? Forgetting to check both conditions (monotonicity and limit to zero) or misinterpreting the results are common mistakes.
9. Is there a visual method to check for monotonicity? Plotting the terms b_n can provide visual confirmation of monotonicity, although this is not a rigorous proof.
Related Articles:
1. The Ratio Test and its Applications: This article compares the Ratio Test with the Alternating Series Test, highlighting their strengths and weaknesses in various contexts.
2. Advanced Techniques for Determining Series Convergence: This article explores more complex techniques for determining series convergence, including the Cauchy condensation test and Dirichlet's test.
3. Error Bounds in Alternating Series Approximations: A detailed explanation of the alternating series estimation theorem and its practical applications with worked examples.
4. Conditional vs. Absolute Convergence: A Comprehensive Guide: This guide explains the difference between conditional and absolute convergence and provides examples to illustrate the concepts.
5. Alternating Series and Power Series: Connections and Applications: This article explores the relationship between alternating series and power series, including applications to Taylor and Maclaurin series.
6. Solving Challenging Alternating Series Problems using the Limit Comparison Test: This article focuses on applying the limit comparison test in conjunction with the alternating series test.
7. The Integral Test and its Relationship to the Alternating Series Test: A comparison of these two tests, highlighting their similarities and differences, particularly in the context of p-series.
8. Visualizing Convergence: Graphical Representations of Series: This article discusses the use of graphical methods to help visualize convergence and divergence of alternating series.
9. Common Mistakes to Avoid When Applying Convergence Tests: A review of common errors made when applying various convergence tests, including the alternating series test, to prevent those mistakes from happening.
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| alternating series test practice problems: AP® Calculus AB & BC Crash Course, 2nd Ed., Book + Online J. Rosebush, Flavia Banu, 2016-10-06 REA's Crash Course® for the AP® Calculus AB & BC Exams - Gets You a Higher Advanced Placement® Score in Less Time 2nd Edition - Updated for the 2017 Exams The REA Crash Course is the top choice for the last-minute studier, or any student who wants a quick refresher on the subject. Are you crunched for time? Have you started studying for your Advanced Placement® Calculus AB & BC exams yet? How will you memorize everything you need to know before the tests? Do you wish there was a fast and easy way to study for the exams and boost your score? If this sounds like you, don't panic. REA's Crash Course for AP® Calculus AB & BC is just what you need. Go with America’s No. 1 quick-review prep for AP® exams to get these outstanding features: Targeted, Focused Review - Study Only What You Need to Know The REA Crash Course is based on an in-depth analysis of the AP® Calculus AB & BC course description outline and actual AP® test questions. It covers only the information tested on the exams, so you can make the most of your valuable study time. Written by experienced AP® Calculus instructors, the targeted review chapters prepare students for the test by only focusing on the topics tested on the AP® Calculus AB & BC exams. Our easy-to-read format gives students a quick but strategic course in AP® Calculus AB & BC and covers functions, graphs, units, derivatives, integrals, and polynomial approximations and series. Expert Test-taking Strategies Our author shares detailed question-level strategies and explain the best way to answer AP® questions you'll find on the exams. By following this expert tips and advice, you can boost your overall point score! Take REA's Practice Exams After studying the material in the Crash Course, go to the online REA Study Center and test what you've learned. Our online practice exams (one for Calculus AB, one for Calculus BC) feature timed testing, detailed explanations of answers, and automatic scoring analysis. Each exam is balanced to include every topic and type of question found on the actual AP® exam, so you know you're studying the smart way. Whether you're cramming for the test at the last minute, looking for an extra edge, or want to study on your own in preparation for the exams - this is the quick-review study guide every AP® Calculus AB & BC student should have. When it’s crunch time and your Advanced Placement® exam is just around the corner, you need REA's Crash Course® for AP® Calculus AB & BC! About the Authors Joan Marie Rosebush teaches calculus courses at the University of Vermont. Ms. Rosebush has taught mathematics to elementary, middle school, high school, and college students. She taught AP® Calculus via satellite television to high school students scattered throughout Vermont. Ms. Rosebush earned her Bachelor of Arts degree in elementary education, with a concentration in mathematics, at the University of New York in Cortland, N.Y. She received her Master's Degree in education from Saint Michael's College, Colchester, Vermont. Flavia Banu graduated from Queens College of the City University of New York with a B.A. in Pure Mathematics and an M.A.in Pure Mathematics in 1997. Ms. Banu was an adjunct professor at Queens College where she taught Algebra and Calculus II. Currently, she teaches mathematics at Bayside High School in Bayside, New York, and coaches the math team for the school. Her favorite course to teach is AP® Calculus because it requires “the most discipline, rigor and creativity.” About Our Editor and Technical Accuracy Checker Stu Schwartz has been teaching mathematics since 1973. For 35 years he taught in the Wissahickon School District, in Ambler, Pennsylvania, specializing in AP® Calculus AB and BC and AP® Statistics. Mr. Schwartz received his B.S. degree in Mathematics from Temple University, Philadelphia. Mr. Schwartz was a 2002 recipient of the Presidential Award for Excellence in Mathematics Teaching and also won the 2007 Outstanding Educator of the Year Award for the Wissahickon School District. Mr. Schwartz’s website, www.mastermathmentor.com, is geared toward helping educators teach AP® Calculus, AP® Statistics, and other math courses. Mr. Schwartz is always looking for ways to provide teachers with new and innovative teaching materials, believing that it should be the goal of every math teacher not only to teach students mathematics, but also to find joy and beauty in math as well. |
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| alternating series test practice problems: Casual Calculus: A Friendly Student Companion (In 3 Volumes) Kenneth Luther, 2022-08-16 Yes, this is another Calculus book. However, it fits in a niche between the two predominant types of such texts. It could be used as a textbook, albeit a streamlined one — it contains exposition on each topic, with an introduction, rationale, train of thought, and solved examples with accompanying suggested exercises. It could be used as a solution guide — because it contains full written solutions to each of the hundreds of exercises posed inside. But its best position is right in between these two extremes. It is best used as a companion to a traditional text or as a refresher — with its conversational tone, its 'get right to it' content structure, and its inclusion of complete solutions to many problems, it is a friendly partner for students who are learning Calculus, either in class or via self-study.Exercises are structured in three sets to force multiple encounters with each topic. Solved examples in the text are accompanied by 'You Try It' problems, which are similar to the solved examples; the students use these to see if they're ready to move forward. Then at the end of the section, there are 'Practice Problems': more problems similar to the 'You Try It' problems, but given all at once. Finally, each section has Challenge Problems — these lean to being equally or a bit more difficult than the others, and they allow students to check on what they've mastered.The goal is to keep the students engaged with the text, and so the writing style is very informal, with attempts at humor along the way. The target audience is STEM students including those in engineering and meteorology programs. |
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| alternating series test practice problems: Casual Calculus: A Friendly Student Companion - Volume 2 Kenneth Luther, 2022-08-16 Yes, this is another Calculus book. However, it fits in a niche between the two predominant types of such texts. It could be used as a textbook, albeit a streamlined one — it contains exposition on each topic, with an introduction, rationale, train of thought, and solved examples with accompanying suggested exercises. It could be used as a solution guide — because it contains full written solutions to each of the hundreds of exercises posed inside. But its best position is right in between these two extremes. It is best used as a companion to a traditional text or as a refresher — with its conversational tone, its 'get right to it' content structure, and its inclusion of complete solutions to many problems, it is a friendly partner for students who are learning Calculus, either in class or via self-study.Exercises are structured in three sets to force multiple encounters with each topic. Solved examples in the text are accompanied by 'You Try It' problems, which are similar to the solved examples; the students use these to see if they're ready to move forward. Then at the end of the section, there are 'Practice Problems': more problems similar to the 'You Try It' problems, but given all at once. Finally, each section has Challenge Problems — these lean to being equally or a bit more difficult than the others, and they allow students to check on what they've mastered.The goal is to keep the students engaged with the text, and so the writing style is very informal, with attempts at humor along the way. The target audience is STEM students including those in engineering and meteorology programs. |
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| alternating series test practice problems: A Problem Book in Real Analysis Asuman G. Aksoy, Mohamed A. Khamsi, 2010-03-10 Education is an admirable thing, but it is well to remember from time to time that nothing worth knowing can be taught. Oscar Wilde, “The Critic as Artist,” 1890. Analysis is a profound subject; it is neither easy to understand nor summarize. However, Real Analysis can be discovered by solving problems. This book aims to give independent students the opportunity to discover Real Analysis by themselves through problem solving. ThedepthandcomplexityofthetheoryofAnalysiscanbeappreciatedbytakingaglimpseatits developmental history. Although Analysis was conceived in the 17th century during the Scienti?c Revolution, it has taken nearly two hundred years to establish its theoretical basis. Kepler, Galileo, Descartes, Fermat, Newton and Leibniz were among those who contributed to its genesis. Deep conceptual changes in Analysis were brought about in the 19th century by Cauchy and Weierstrass. Furthermore, modern concepts such as open and closed sets were introduced in the 1900s. Today nearly every undergraduate mathematics program requires at least one semester of Real Analysis. Often, students consider this course to be the most challenging or even intimidating of all their mathematics major requirements. The primary goal of this book is to alleviate those concerns by systematically solving the problems related to the core concepts of most analysis courses. In doing so, we hope that learning analysis becomes less taxing and thereby more satisfying. |
| alternating series test practice problems: Partial Differential Equations and Boundary-Value Problems with Applications Mark A. Pinsky, 2011 Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems--rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate throughout the text. The notions of steady-state solution of closely related stationary solutions are developed for the heat equation; applications to the study of heat flow in the earth are presented. The problem of the vibrating string is studied in detail both in the Fourier transform setting and from the viewpoint of the explicit representation (d'Alembert formula). Additional chapters include the numerical analysis of solutions and the method of Green's functions for solutions of partial differential equations. The exposition also includes asymptotic methods (Laplace transform and stationary phase). With more than 200 working examples and 700 exercises (more than 450 with answers), the book is suitable for an undergraduate course in partial differential equations. |
| alternating series test practice problems: Calculus Kenneth Kuttler, 2011 This is a book on single variable calculus including most of the important applications of calculus. It also includes proofs of all theorems presented, either in the text itself, or in an appendix. It also contains an introduction to vectors and vector products which is developed further in Volume 2. While the book does include all the proofs of the theorems, many of the applications are presented more simply and less formally than is often the case in similar titles. Supplementary materials are available upon request for all instructors who adopt this book as a course text. Please send your request to sales@wspc.com. This book is also available as a set with Volume 2: CALCULUS: Theory and Applications. |
| alternating series test practice problems: Advanced Calculus John Srdjan Petrovic, 2013-11-01 Suitable for a one- or two-semester course, Advanced Calculus: Theory and Practice expands on the material covered in elementary calculus and presents this material in a rigorous manner. The text improves students’ problem-solving and proof-writing skills, familiarizes them with the historical development of calculus concepts, and helps them understand the connections among different topics. The book takes a motivating approach that makes ideas less abstract to students. It explains how various topics in calculus may seem unrelated but in reality have common roots. Emphasizing historical perspectives, the text gives students a glimpse into the development of calculus and its ideas from the age of Newton and Leibniz to the twentieth century. Nearly 300 examples lead to important theorems as well as help students develop the necessary skills to closely examine the theorems. Proofs are also presented in an accessible way to students. By strengthening skills gained through elementary calculus, this textbook leads students toward mastering calculus techniques. It will help them succeed in their future mathematical or engineering studies. |
| alternating series test practice problems: Calculus Gerald L. Bradley, 1995 |
| alternating series test practice problems: Differential Equations Steven G. Krantz, 2015-10-07 Differential Equations: Theory, Technique, and Practice with Boundary Value Problems presents classical ideas and cutting-edge techniques for a contemporary, undergraduate-level, one- or two-semester course on ordinary differential equations. Authored by a widely respected researcher and teacher, the text covers standard topics such as partial diff |
| alternating series test practice problems: Methods of Solving Sequence and Series Problems Ellina Grigorieva, 2016-12-09 This book aims to dispel the mystery and fear experienced by students surrounding sequences, series, convergence, and their applications. The author, an accomplished female mathematician, achieves this by taking a problem solving approach, starting with fascinating problems and solving them step by step with clear explanations and illuminating diagrams. The reader will find the problems interesting, unusual, and fun, yet solved with the rigor expected in a competition. Some problems are taken directly from mathematics competitions, with the name and year of the exam provided for reference. Proof techniques are emphasized, with a variety of methods presented. The text aims to expand the mind of the reader by often presenting multiple ways to attack the same problem, as well as drawing connections with different fields of mathematics. Intuitive and visual arguments are presented alongside technical proofs to provide a well-rounded methodology. With nearly 300 problems including hints, answers, and solutions, Methods of Solving Sequences and Series Problems is an ideal resource for those learning calculus, preparing for mathematics competitions, or just looking for a worthwhile challenge. It can also be used by faculty who are looking for interesting and insightful problems that are not commonly found in other textbooks. |
| alternating series test practice problems: Berkeley Problems in Mathematics Paulo Ney de Souza, Jorge-Nuno Silva, 2004-01-08 This book collects approximately nine hundred problems that have appeared on the preliminary exams in Berkeley over the last twenty years. It is an invaluable source of problems and solutions. Readers who work through this book will develop problem solving skills in such areas as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. |
| alternating series test practice problems: AP Calculus Premium, 2025: Prep Book with 12 Practice Tests + Comprehensive Review + Online Practice David Bock, Dennis Donovan, Shirley O. Hockett, 2024-07-02 Be prepared for exam day with Barron’s. Trusted content from AP experts! Barron’s AP Calculus Premium, 2025 includes in‑depth content review and practice for the AB and BC exams. It’s the only book you’ll need to be prepared for exam day. Written by Experienced Educators Learn from Barron’s‑‑all content is written and reviewed by AP experts Build your understanding with comprehensive review tailored to the most recent exams Get a leg up with tips, strategies, and study advice for exam day‑‑it’s like having a trusted tutor by your side Be Confident on Exam Day Sharpen your test‑taking skills with 12 full‑length practice tests‑‑3 AB practice tests and 3 BC practice tests in the book, including one diagnostic test each for AB and BC to target your studying‑‑and 3 more AB practice tests and 3 more BC practice tests online–plus detailed answer explanations for all questions Strengthen your knowledge with in‑depth review covering all units on the AP Calculus AB and BC exams Reinforce your learning with dozens of examples and detailed solutions, plus a series of multiple‑choice practice questions and answer explanations, within each chapter Enhance your problem‑solving skills by working through a chapter filled with multiple‑choice questions on a variety of tested topics and a chapter devoted to free‑response practice exercises Robust Online Practice Continue your practice with 3 full‑length AB practice tests and 3 full‑length BC practice tests on Barron’s Online Learning Hub Simulate the exam experience with a timed test option Deepen your understanding with detailed answer explanations and expert advice Gain confidence with scoring to check your learning progress |
| alternating series test practice problems: Contemporary Calculus III Dale Hoffman, 2012-01-23 This is a textbook for 3rd quarter calculus covering the three main topics of (1) calculus with polar coordinates and parametric equations, (2) infinite series, and (3) vectors in 3D. It has explanations, examples, worked solutions, problem sets and answers. It has been reviewed by calculus instructors and class-tested by them and the author. Besides technique practice and applications of the techniques, the examples and problem sets are also designed to help students develop a visual and conceptual understanding of the main ideas. The exposition and problem sets have been highly rated by reviewers. |
| alternating series test practice problems: Schaum's Outline of Calculus, 6th Edition Frank Ayres, Elliott Mendelson, 2012-11-16 Tough Test Questions? Missed Lectures? Not Enough Time? Fortunately, there's Schaum's. This all-in-one-package includes more than 1,100 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 30 detailed videos featuring Math instructors who explain how to solve the most commonly tested problems--it's just like having your own virtual tutor! You’ll find everything you need to build confidence, skills, and knowledge for the highest score possible. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. This Schaum's Outline gives you 1,105 fully solved problems Concise explanations of all calculus concepts Expert tips on using the graphing calculator Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time--and get your best test scores! |
| alternating series test practice problems: Cracking the A. P. Calculus David Kahn, 1998-01-15 THE BOOK THAT GETS YOU RESULTS *Includes two full-length AP Calculus practice tests, one each for the AB & BC exams. *Sharpen your skills with more than 900 practice questions. *Review the essential calculus covered on the exam. WE KNOW THE AP CALCULUS AB & BC EXAMS The experts at The Princeton Review study the AP Calculus exam and other standardized tests each year to make sure you get the most up-to-date, thoroughly researched books possible. WE KNOW STUDENTS Each year we help more than two million students score high with our courses, bestselling books, and award-winning software. WE GET RESULTS Students who take our courses for the SAT, GRE, LSAT, and many other tests see score improvements that have been verified by independent accounting firms. The proven techniques we teach in our courses are in this book. AND IF IT'S ON THE AP CALCULUS EXAM, IT'S IN THIS BOOK We don't try to teach you everything there is to know about calculus-only the facts and techniques you'll need to know to score high on the Advanced Placement exam. There's a big difference. In Cracking the AP Calculus AB & BC, 1998-1999 Edition, you will learn to think like the test-makers and: *Review and practice the calculus concepts that are covered on the exam *Score higher by mastering a few essential problem-solving techniques *Immediately recognize problem types and recall the techniques that are needed to solve them *Memorize important formulas so you won't have to rely on your calculator *Become a test-taking expert by practicing on the more than 900 problems in this book Practice your skills on the full-length sample tests inside (one each for boththe AB and BC exams). The questions are just like the ones you'll see on the actual AP Calculus exam, and we fully explain every answer. |
| alternating series test practice problems: Prime Obsession John Derbyshire, 2003-04-15 In August 1859 Bernhard Riemann, a little-known 32-year old mathematician, presented a paper to the Berlin Academy titled: On the Number of Prime Numbers Less Than a Given Quantity. In the middle of that paper, Riemann made an incidental remark †a guess, a hypothesis. What he tossed out to the assembled mathematicians that day has proven to be almost cruelly compelling to countless scholars in the ensuing years. Today, after 150 years of careful research and exhaustive study, the question remains. Is the hypothesis true or false? Riemann's basic inquiry, the primary topic of his paper, concerned a straightforward but nevertheless important matter of arithmetic †defining a precise formula to track and identify the occurrence of prime numbers. But it is that incidental remark †the Riemann Hypothesis †that is the truly astonishing legacy of his 1859 paper. Because Riemann was able to see beyond the pattern of the primes to discern traces of something mysterious and mathematically elegant shrouded in the shadows †subtle variations in the distribution of those prime numbers. Brilliant for its clarity, astounding for its potential consequences, the Hypothesis took on enormous importance in mathematics. Indeed, the successful solution to this puzzle would herald a revolution in prime number theory. Proving or disproving it became the greatest challenge of the age. It has become clear that the Riemann Hypothesis, whose resolution seems to hang tantalizingly just beyond our grasp, holds the key to a variety of scientific and mathematical investigations. The making and breaking of modern codes, which depend on the properties of the prime numbers, have roots in the Hypothesis. In a series of extraordinary developments during the 1970s, it emerged that even the physics of the atomic nucleus is connected in ways not yet fully understood to this strange conundrum. Hunting down the solution to the Riemann Hypothesis has become an obsession for many †the veritable great white whale of mathematical research. Yet despite determined efforts by generations of mathematicians, the Riemann Hypothesis defies resolution. Alternating passages of extraordinarily lucid mathematical exposition with chapters of elegantly composed biography and history, Prime Obsession is a fascinating and fluent account of an epic mathematical mystery that continues to challenge and excite the world. Posited a century and a half ago, the Riemann Hypothesis is an intellectual feast for the cognoscenti and the curious alike. Not just a story of numbers and calculations, Prime Obsession is the engrossing tale of a relentless hunt for an elusive proof †and those who have been consumed by it. |
| alternating series test practice problems: Advanced Calculus (Revised Edition) Lynn Harold Loomis, Shlomo Zvi Sternberg, 2014-02-26 An authorised reissue of the long out of print classic textbook, Advanced Calculus by the late Dr Lynn Loomis and Dr Shlomo Sternberg both of Harvard University has been a revered but hard to find textbook for the advanced calculus course for decades.This book is based on an honors course in advanced calculus that the authors gave in the 1960's. The foundational material, presented in the unstarred sections of Chapters 1 through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in any one year. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis.The prerequisites are a good grounding in the calculus of one variable from a mathematically rigorous point of view, together with some acquaintance with linear algebra. The reader should be familiar with limit and continuity type arguments and have a certain amount of mathematical sophistication. As possible introductory texts, we mention Differential and Integral Calculus by R Courant, Calculus by T Apostol, Calculus by M Spivak, and Pure Mathematics by G Hardy. The reader should also have some experience with partial derivatives.In overall plan the book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds. |
| alternating series test practice problems: Exploring the Infinite Jennifer Brooks, 2016-11-30 Exploring the Infinite addresses the trend toward a combined transition course and introduction to analysis course. It guides the reader through the processes of abstraction and log- ical argumentation, to make the transition from student of mathematics to practitioner of mathematics. This requires more than knowledge of the definitions of mathematical structures, elementary logic, and standard proof techniques. The student focused on only these will develop little more than the ability to identify a number of proof templates and to apply them in predictable ways to standard problems. This book aims to do something more; it aims to help readers learn to explore mathematical situations, to make conjectures, and only then to apply methods of proof. Practitioners of mathematics must do all of these things. The chapters of this text are divided into two parts. Part I serves as an introduction to proof and abstract mathematics and aims to prepare the reader for advanced course work in all areas of mathematics. It thus includes all the standard material from a transition to proof course. Part II constitutes an introduction to the basic concepts of analysis, including limits of sequences of real numbers and of functions, infinite series, the structure of the real line, and continuous functions. Features Two part text for the combined transition and analysis course New approach focuses on exploration and creative thought Emphasizes the limit and sequences Introduces programming skills to explore concepts in analysis Emphasis in on developing mathematical thought Exploration problems expand more traditional exercise sets |
| alternating series test practice problems: The American Mathematical Monthly , 1918 Includes section Recent publications. |
ALTERNATING | English meaning - Cambridge Dictionary
ALTERNATING definition: 1. with first one thing, then another thing, and then the first thing again: 2. happening every…. Learn more.
ALTERNATING Definition & Meaning - Merriam-Webster
The meaning of ALTERNATING is occurring by turns or in succession. How to use alternating in a sentence.
Alternating current - Wikipedia
Alternating current (AC) is an electric current that periodically reverses direction and changes its magnitude continuously with time, in contrast to direct current (DC), which flows only in one …
Alternating - definition of alternating by The Free Dictionary
To pass back and forth from one state, action, or place to another: alternated between happiness and depression. 3. Electricity To reverse direction at regular intervals in a circuit. 1. To do or …
ALTERNATING definition and meaning | Collins English Dictionary
their alternating periods of excitement and repose an imaginative novel, with alternating chapters presenting each partner's point of view. alternating lines of red and yellow
alternating, n. meanings, etymology and more - Oxford English …
There is one meaning in OED's entry for the noun alternating. See ‘Meaning & use’ for definition, usage, and quotation evidence.
What does Alternating mean? - Definitions.net
Alternating generally refers to a process or pattern that switches between two or more things in a regular or predictable sequence. It indicates an action of changing back and forth, or swinging …
alternating | English Definition & Examples | Ludwig
The word "alternating" is correct and usable in written English You can use it to describe something that occurs or is done in turns, one after the other. For example: "The teacher …
Alternating - Definition, Meaning & Synonyms - Vocabulary.com
Anything that alternates can be described as alternating. An easy way to remember the meaning of alternating to think of the alter inside it: to alter means "to change," so if something is …
alternating - WordReference.com Dictionary of English
to change back and forth between conditions, states, actions, etc.: He alternates between hope and despair. to take turns: My sister and I alternated in doing the dishes. Electricity to reverse …
ALTERNATING | English meaning - Cambridge Dictionary
ALTERNATING definition: 1. with first one thing, then another thing, and then the first thing again: 2. happening every…. Learn more.
ALTERNATING Definition & Meaning - Merriam-Webster
The meaning of ALTERNATING is occurring by turns or in succession. How to use alternating in a sentence.
Alternating current - Wikipedia
Alternating current (AC) is an electric current that periodically reverses direction and changes its magnitude continuously with time, in contrast to direct current (DC), which flows only in one …
Alternating - definition of alternating by The Free Dictionary
To pass back and forth from one state, action, or place to another: alternated between happiness and depression. 3. Electricity To reverse direction at regular intervals in a circuit. 1. To do or …
ALTERNATING definition and meaning | Collins English Dictionary
their alternating periods of excitement and repose an imaginative novel, with alternating chapters presenting each partner's point of view. alternating lines of red and yellow
alternating, n. meanings, etymology and more - Oxford English …
There is one meaning in OED's entry for the noun alternating. See ‘Meaning & use’ for definition, usage, and quotation evidence.
What does Alternating mean? - Definitions.net
Alternating generally refers to a process or pattern that switches between two or more things in a regular or predictable sequence. It indicates an action of changing back and forth, or swinging …
alternating | English Definition & Examples | Ludwig
The word "alternating" is correct and usable in written English You can use it to describe something that occurs or is done in turns, one after the other. For example: "The teacher …
Alternating - Definition, Meaning & Synonyms - Vocabulary.com
Anything that alternates can be described as alternating. An easy way to remember the meaning of alternating to think of the alter inside it: to alter means "to change," so if something is …
alternating - WordReference.com Dictionary of English
to change back and forth between conditions, states, actions, etc.: He alternates between hope and despair. to take turns: My sister and I alternated in doing the dishes. Electricity to reverse …
