Numbers, Sequences and Series

Author: Keith E. Hirst

Publisher: Butterworth-Heinemann

ISBN: 0340610433

Category: Mathematics

Page: 198

View: 2055

Concerned with the logical foundations of number systems from integers to complex numbers.

Problems in Mathematical Analysis: Real numbers, sequences, and series

Author: Wiesława J. Kaczor,Wiesława Kaczor,Maria T. Nowak

Publisher: American Mathematical Soc.

ISBN: 0821820508

Category: Mathematics

Page: 380

View: 2378

We learn by doing. We learn mathematics by doing problems. This book is the first volume of a series of books of problems in mathematical analysis. It is mainly intended for students studying the basic principles of analysis. However, given its organization, level, and selection of problems, it would also be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. The volume is also suitable for self-study. Each section of the book begins with relatively simple exercises, yet may also contain quite challenging problems. Very often several consecutive exercises are concerned with different aspects of one mathematical problem or theorem.This presentation of material is designed to help student comprehension and to encourage them to ask their own questions and to start research. The collection of problems in the book is also intended to help teachers who wish to incorporate the problems into lectures. Solutions for all the problems are provided. The book covers three topics: real numbers, sequences, and series, and is divided into two parts: exercises and/or problems, and solutions. Specific topics covered in this volume include the following: basic properties of real numbers, continued fractions, monotonic sequences, limits of sequences, Stolz's theorem, summation of series, tests for convergence, double series, arrangement of series, Cauchy product, and infinite products. Also available from the AMS are ""Problems in Mathematical Analysis II"" and ""Problems in Analysis III"" in the ""Student Mathematical Library"" series.

Infinite Sequences and Series

Author: Konrad Knopp

Publisher: Courier Corporation

ISBN: 0486152049

Category: Mathematics

Page: 208

View: 2381

Careful presentation of fundamentals of the theory by one of the finest modern expositors of higher mathematics. Covers functions of real and complex variables, arbitrary and null sequences, convergence and divergence, Cauchy's limit theorem, more.

Problems in Mathematical Analysis: Real numbers, sequences, and series

Author: Wiesława J. Kaczor,Wiesława Kaczor,Maria T. Nowak

Publisher: American Mathematical Soc.

ISBN: 0821820508

Category: Mathematics

Page: 380

View: 1138

We learn by doing. We learn mathematics by doing problems. This book is the first volume of a series of books of problems in mathematical analysis. It is mainly intended for students studying the basic principles of analysis. However, given its organization, level, and selection of problems, it would also be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. The volume is also suitable for self-study. Each section of the book begins with relatively simple exercises, yet may also contain quite challenging problems. Very often several consecutive exercises are concerned with different aspects of one mathematical problem or theorem.This presentation of material is designed to help student comprehension and to encourage them to ask their own questions and to start research. The collection of problems in the book is also intended to help teachers who wish to incorporate the problems into lectures. Solutions for all the problems are provided. The book covers three topics: real numbers, sequences, and series, and is divided into two parts: exercises and/or problems, and solutions. Specific topics covered in this volume include the following: basic properties of real numbers, continued fractions, monotonic sequences, limits of sequences, Stolz's theorem, summation of series, tests for convergence, double series, arrangement of series, Cauchy product, and infinite products. Also available from the AMS are ""Problems in Mathematical Analysis II"" and ""Problems in Analysis III"" in the ""Student Mathematical Library"" series.

Methods of Solving Sequence and Series Problems

Author: Ellina Grigorieva

Publisher: Birkhäuser

ISBN: 3319456865

Category: Mathematics

Page: 281

View: 3062

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.

Irrational Numbers and Their Representation by Sequences and Series

Author: Henry Parker Manning

Publisher: Createspace Independent Publishing Platform

ISBN: 9781534890169

Category:

Page: 130

View: 8938

Dr. Manning's book on irrational numbers contains a presentation in a simple form of another field of mathematical inquiry, such as is also eminently_ suited for placing in the hands of the ordinary schoolmaster. We have decided that the geometry of proportion shall be taught to schoolboys without reference to irrational quantities, but we have not yet eliminated a spirit of reckless extravagance in the quite unnecessary use of infinite series, often with total disregard for their convergency. In Dr. Manning's treatment an irrational number is defined as forming a point of separation between rational numbers of two classes, the numbers of one class being less than those of the other. This definition appears to involve the assumption (pp. 7, 10, &c.) that the point of separation is unique, in other words, that there cannot be two irrational numbers which have not some rational number separating them. Perhaps this assumption may be regarded as a definition of equality of irrational numbers; in any case, the inquiring reader would find it necessary to examine more fully the references to Dedekind's and Cantor's writings given on p. 56. Once the assumption or definition is made, the representation of numbers by sequences readily follows. The theory of limits is discussed on p. 57, and in the following chapter the notion of a sequence is shown to give rise to that of a series. The remaining portion of the book is mainly devoted to the study of convergence, and includes the well-known multiplication theorem and applications to the still better-known binomial and exponential series. -Nature, Vol. 75

Elements of Real Analysis

Author: David A. Sprecher

Publisher: Courier Corporation

ISBN: 0486153258

Category: Mathematics

Page: 368

View: 1003

Classic text explores intermediate steps between basics of calculus and ultimate stage of mathematics — abstraction and generalization. Covers fundamental concepts, real number system, point sets, functions of a real variable, Fourier series, more. Over 500 exercises.

Real Analysis via Sequences and Series

Author: Charles H.C. Little,Kee L. Teo,Bruce van Brunt

Publisher: Springer

ISBN: 1493926519

Category: Mathematics

Page: 476

View: 1135

This text gives a rigorous treatment of the foundations of calculus. In contrast to more traditional approaches, infinite sequences and series are placed at the forefront. The approach taken has not only the merit of simplicity, but students are well placed to understand and appreciate more sophisticated concepts in advanced mathematics. The authors mitigate potential difficulties in mastering the material by motivating definitions, results and proofs. Simple examples are provided to illustrate new material and exercises are included at the end of most sections. Noteworthy topics include: an extensive discussion of convergence tests for infinite series, Wallis’s formula and Stirling’s formula, proofs of the irrationality of π and e and a treatment of Newton’s method as a special instance of finding fixed points of iterated functions.

Problems and Theorems in Analysis I

Series. Integral Calculus. Theory of Functions

Author: George Polya,Gabor Szegö

Publisher: Springer Science & Business Media

ISBN: 9783540636403

Category: Mathematics

Page: 393

View: 9983

From the reviews: "The work is one of the real classics of this century; it has had much influence on teaching, on research in several branches of hard analysis, particularly complex function theory, and it has been an essential indispensable source book for those seriously interested in mathematical problems." Bulletin of the American Mathematical Society

Theory and Application of Infinite Series

Author: Konrad Knopp

Publisher: Courier Corporation

ISBN: 0486318613

Category: Mathematics

Page: 592

View: 5381

Unusually clear and interesting classic covers real numbers and sequences, foundations of the theory of infinite series and development of the theory (series of valuable terms, Euler's summation formula, asymptotic expansions, other topics). Includes exercises.

Fibonacci and Lucas Numbers with Applications

Author: Thomas Koshy

Publisher: John Wiley & Sons

ISBN: 1118031318

Category: Mathematics

Page: 672

View: 8077

The first comprehensive survey of mathematics' most fascinatingnumber sequences Fibonacci and Lucas numbers have intrigued amateur and professionalmathematicians for centuries. This volume represents the firstattempt to compile a definitive history and authoritative analysisof these famous integer sequences, complete with a wealth ofexciting applications, enlightening examples, and fun exercisesthat offer numerous opportunities for exploration andexperimentation. The author has assembled a myriad of fascinating properties of bothFibonacci and Lucas numbers-as developed by a wide range ofsources-and catalogued their applications in a multitude of widelyvaried disciplines such as art, stock market investing,engineering, and neurophysiology. Most of the engaging anddelightful material here is easily accessible to college and evenhigh school students, though advanced material is included tochallenge more sophisticated Fibonacci enthusiasts. A historicalsurvey of the development of Fibonacci and Lucas numbers,biographical sketches of intriguing personalities involved indeveloping the subject, and illustrative examples round out thisthorough and amusing survey. Most chapters conclude with numericand theoretical exercises that do not rely on long and tediousproofs of theorems. Highlights include: * Balanced blend of theory and real-world applications * Excellent reference material for student reports andprojects * User-friendly, informal, and entertaining writing style * Historical interjections and short biographies that add a richerperspective to the topic * Reference sections providing important symbols, problemsolutions, and fundamental properties from the theory of numbersand matrices Fibonacci and Lucas Numbers with Applications providesmathematicians with a wealth of reference material in oneconvenient volume and presents an in-depth and entertainingresource for enthusiasts at every level and from any background.

Limits, Limits Everywhere

The Tools of Mathematical Analysis

Author: David Applebaum

Publisher: Oxford University Press

ISBN: 0199640084

Category: Mathematics

Page: 200

View: 4068

An account of elementary real analysis positioned between a popular mathematics book and a first year college or university text. This book doesn't assume knowledge of calculus and, instead, the emphasis is on the application of analysis to number theory.

Problems in Mathematical Analysis: Continuity and differentiation

Author: Wiesława J. Kaczor,Maria T. Nowak

Publisher: American Mathematical Soc.

ISBN: 0821820516

Category: Mathematics

Page: 398

View: 4107

We learn by doing. We learn mathematics by doing problems. And we learn more mathematics by doing more problems. This is the sequel to Problems in Mathematical Analysis I (Volume 4 in the Student Mathematical Library series). If you want to hone your understanding of continuous and differentiable functions, this book contains hundreds of problems to help you do so. The emphasis here is on real functions of a single variable. The book is mainly geared toward students studying the basic principles of analysis. However, given its selection of problems, organization, and level, it would be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. It is also suitable for self-study. The presentation of the material is designed to help student comprehension, to encourage them to ask their own questions, and to start research. The collection of problems will also help teachers who wish to incorporate problems into their lectures. The problems are grouped into sections according to the methods of solution. Solutions for the problems are provided.

Mathematical Mysteries

The Beauty and Magic of Numbers

Author: Calvin C. Clawson

Publisher: Springer

ISBN: 1489960805

Category: Mathematics

Page: 314

View: 6104

A meditation on the beauty and meaning of numbers, exploring mathematical equations, describing some of the mathematical discoveries of the past millennia, and pondering philosophical questions about the relation of numbers to the universe.

Foundations of Mathematical Analysis

Author: Saminathan Ponnusamy

Publisher: Springer Science & Business Media

ISBN: 0817682929

Category: Mathematics

Page: 570

View: 3189

Mathematical analysis is fundamental to the undergraduate curriculum not only because it is the stepping stone for the study of advanced analysis, but also because of its applications to other branches of mathematics, physics, and engineering at both the undergraduate and graduate levels. This self-contained textbook consists of eleven chapters, which are further divided into sections and subsections. Each section includes a careful selection of special topics covered that will serve to illustrate the scope and power of various methods in real analysis. The exposition is developed with thorough explanations, motivating examples, exercises, and illustrations conveying geometric intuition in a pleasant and informal style to help readers grasp difficult concepts. Foundations of Mathematical Analysis is intended for undergraduate students and beginning graduate students interested in a fundamental introduction to the subject. It may be used in the classroom or as a self-study guide without any required prerequisites.

The Real Numbers and Real Analysis

Author: Ethan D. Bloch

Publisher: Springer Science & Business Media

ISBN: 0387721762

Category: Mathematics

Page: 554

View: 4151

This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. It is organized in a distinctive, flexible way that would make it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus. The Real Numbers and Real Analysis will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.

Introduction to Real Analysis

Author: Manfred Stoll

Publisher: Addison-Wesley

ISBN: 9780321046253

Category: Mathematics

Page: 550

View: 8479

This text is a single variable real analysis text, designed for the one-year course at the junior, senior, or beginning graduate level. It provides a rigorous and comprehensive treatment of the theoretical concepts of analysis. The book contains most of the topics covered in a text of this nature, but it also includes many topics not normally encountered in comparable texts. These include the Riemann-Stieltjes integral, the Lebesgue integral, Fourier series, the Weiestrass approximation theorem, and an introduction to normal linear spaces. The Real Number System; Sequence Of Real Numbers; Structure Of Point Sets; Limits And Continuity; Differentiation; The Riemann And Riemann-Stieltjes Integral; Series of Real Numbers; Sequences And Series Of Functions; Orthogonal Functions And Fourier Series; Lebesgue Measure And Integration; Logic and Proofs; Propositions and Connectives For all readers interested in real analysis.

A Primer on Number Sequences

Author: S. Shirali

Publisher: Universities Press

ISBN: 9788173713699

Category: Sequences (Mathematics)

Page: 251

View: 5058