The purpose of this book is to isolate and draw attention to the most important problem-solving techniques typically encountered in undergradu ate mathematics and to illustrate their use by interesting examples and problems not easily found in other sources. Each section features a single idea, the power and versatility of which is demonstrated in the examples and reinforced in the problems. The book serves as an introduction and guide to the problems literature (e. g. , as found in the problems sections of undergraduate mathematics journals) and as an easily accessed reference of essential knowledge for students and teachers of mathematics. The book is both an anthology of problems and a manual of instruction. It contains over 700 problems, over one-third of which are worked in detail. Each problem is chosen for its natural appeal and beauty, but primarily to provide the context for illustrating a given problem-solving method. The aim throughout is to show how a basic set of simple techniques can be applied in diverse ways to solve an enormous variety of problems. Whenever possible, problems within sections are chosen to cut across expected course boundaries and to thereby strengthen the evidence that a single intuition is capable of broad application. Each section concludes with "Additional Examples" that point to other contexts where the technique is appropriate.
Inhaltsverzeichnis
1. Heuristics. - 1. 1. Search for a Pattern. - 1. 2. Draw a Figure. - 1. 3. Formulate an Equivalent Problem. - 1. 4. Modify the Problem. - 1. 5. Choose Effective Notation. - 1. 6. Exploit Symmetry. - 1. 7. Divide into Cases. - 1. 8. Work Backward. - 1. 9. Argue by Contradiction. - 1. 10. Pursue Parity. - 1. 11. Consider Extreme Cases. - 1. 12. Generalize. - 2. Two Important Principles: Induction and Pigeonhole. - 2. 1. Induction: Build on P(k). - 2. 2. Induction: Set Up P(k + 1). - 2. 3. Strong Induction. - 2. 4. Induction and Generalization. - 2. 5. Recursion. - 2. 6. Pigeonhole Principle. - 3. Arithmetic. - 3. 1. Greatest Common Divisor. - 3. 2. Modular Arithmetic. - 3. 3. Unique Factorization. - 3. 4. Positional Notation. - 3. 5. Arithmetic of Complex Numbers. - 4. Algebra. - 4. 1. Algebraic Identities. - 4. 2. Unique Factorization of Polynomials. - 4. 3. The Identity Theorem. - 4. 4. Abstract Algebra. - 5. Summation of Series. - 5. 1. Binomial Coefficients. - 5. 2. Geometric Series. - 5. 3. Telescoping Series. - 5. 4. Power Series. - 6. Intermediate Real Analysis. - 6. 1. Continuous Functions. - 6. 2. The Intermediate-Value Theorem. - 6. 3. The Derivative. - 6. 4. The Extreme-Value Theorem. - 6. 5. Rolle s Theorem. - 6. 6. The Mean Value Theorem. - 6. 7. L Hôpital s Rule. - 6. 8. The Integral. - 6. 9. The Fundamental Theorem. - 7. Inequalities. - 7. 1. Basic Inequality Properties. - 7. 2. Arithmetic-Mean-Geometric-Mean Inequality. - 7. 3. Cauchy-Schwarz Inequality. - 7. 4. Functional Considerations. - 7. 5. Inequalities by Series. - 7. 6. The Squeeze Principle. - 8. Geometry. - 8. 1. Classical Plane Geometry. - 8. 2. Analytic Geometry. - 8. 3. Vector Geometry. - 8. 4. Complex Numbers in Geometry. - Glossary of Symbols and Definitions. - Sources.
