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This book is an outgrowth of a collection of sixty-two problems offered in the The American Mathematical Monthly (AMM) the author has worked over the last two decades. Each selected problem has a central theme, contains gems of sophisticated ideas connected to important current research, and opens new vistas in the understanding of mathematics. The AMM problem section provides one of the most challenging and interesting problem sections among the various journals and online sources currently available. The published problems and solutions have become a treasure trove rife with mathematical gems. The author presents either his published solution in the AMM or an alternative solution to the published one to present and develop problem-solving techniques. A rich glossary of important theorems and formulas is included for easy reference. The reader may regard this book as a starter set for AMM problems, providing a jumping of point to new ideas, and extending their personal lexicon of problems and solutions. This collection is intended to encourage the reader to move away from routine exercises toward creative solutions, as well as offering the reader a systematic illustration of how to organize the transition from problem solving to exploring, investigating and discovering new results. This book is an outgrowth of a collection of sixty-two problems offered in the The American Mathematical Monthly (AMM) the author has worked over the last two decades. Cover 1 Half Title 2 Title Page 4 Copyright Page 5 Contents 6 Preface 8 1. Limits 12 1.1. A rational recurrence 12 1.2. Asymptotic behavior for a Pólya type recurrence 15 1.3. A nonlinear recurrence with Fibonacci exponents 20 1.4. Rate of convergence for an integral 23 1.5. How closely does this sum approximate the integral? 26 1.6. An arctangent series 30 1.7. Geometric mean rates 32 1.8. Limits of the weighted mean ratio 35 1.9. Limits of mean recurrences 38 1.10. A disguised half-angle iteration 43 1.11. Nested radicals and generalized Fibonacci numbers 47 1.12. A limit involving arctangent 51 1.13. Summing to the double factorials 54 1.14. A fractional part sum with Euler's constant 58 1.15. A Putnam/Monthly limit problem 61 2. Infinite Series 64 2.1. Wilf wants us thinking rationally 64 2.2. Old wine in a new bottle 68 2.3. Another Euler sum 81 2.4. Reciprocal Catalan number sums 86 2.5. Catalan generating function 90 2.6. A series with log and harmonic numbers 98 2.7. A nonlinear harmonic sum 102 2.8. A series involving Riemann zeta values 106 2.9. Abel theorem continued 110 2.10. A convergence test 114 2.11. A power series with an exponential tail 117 2.12. An in nite matrix product 121 3. Integrations 126 3.1. A Lobachevsky integral 126 3.2. Two log gamma integrals 130 3.3. Short gamma products with simple values 134 3.4. Evaluate an integral by Feynman's way 138 3.5. Three ways to evaluate a log-sine integral 142 3.6. A log-product integral 148 3.7. A Putnam problem beyond 150 3.8. A surface integral with many faces 154 3.9. Evaluate a de nite integral by the gamma function 159 3.10. Digamma via a double integral 162 3.11. Another double integral 166 3.12. An integral with log and arctangent 172 3.13. Another integral with log and arctangent 177 3.14. An orthonormal function sequence 187 3.15. An definite integral Quickie 190 4. Inequalities 192 4.1. An inequality from Klamkin 192 4.2. Knuth's exponential inequality 198 4.3. Tight bounds for the normal distribution 204 4.4. An inequality due to Knopp 208 4.5. A discrete inequality by integral 215 4.6. An inequality by power series 218 4.7. An inequality by differential equation 223 4.8. Bounds for a reciprocal of log sum 228 4.9. Log-concavity of a partial binomial sum 233 4.10. A bound of divisor sums related to the Riemann 236 5. Monthly Miniatures 246 5.1. Value defined by an integral 246 5.2. Another mean value theorem 250 5.3. The product of derivatives by Darboux's theorem 254 5.4. An integral-derivative inequality 257 5.5. Eigenvalues of a (0; 1)-matrix 261 5.6. A matrix of secants 266 5.7. A gcd-weighted trigonometric sum 274 5.8. A lcm-sum weighted Dirichlet series 278 5.9. An in nite sum-product identity 284 5.10. A polynomial zero identity 291 A. List of Problems 296 B. Glossary 300 Bibliography 314 Index 322 Limits;,rational,recurrence;,Asymptotic,behavior;,Arctangent;,Fibonacci;,Infinite,Series;,Euler;,Catalan;,Riemann;,Integrations;,Lobachevsky Limits,rational recurrence,Asymptotic behavior,Arctangent,Fibonacci,Infinite Series,Euler,Catalan,Riemann,Integrations,Lobachevsky