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Monthly Problem Gems [Pehme köide]

(Department of Mathematics, Christopher Newport University.)
  • Formaat: Paperback / softback, 319 pages, kõrgus x laius: 234x156 mm, kaal: 471 g, 7 Halftones, black and white; 7 Illustrations, black and white
  • Ilmumisaeg: 06-Jul-2021
  • Kirjastus: CRC Press
  • ISBN-10: 0367766779
  • ISBN-13: 9780367766771
Teised raamatud teemal:
Monthly Problem GemsSuurem pilt
  • Formaat: Paperback / softback, 319 pages, kõrgus x laius: 234x156 mm, kaal: 471 g, 7 Halftones, black and white; 7 Illustrations, black and white
  • Ilmumisaeg: 06-Jul-2021
  • Kirjastus: CRC Press
  • ISBN-10: 0367766779
  • ISBN-13: 9780367766771
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780367766771.html
Märksõnad:

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.



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.

Preface vii
1 Limits
1(52)
1.1 A rational recurrence
1(3)
1.2 Asymptotic behavior for a Polya type recurrence
4(5)
1.3 A nonlinear recurrence with Fibonacci exponents
9(3)
1.4 Rate of convergence for an integral
12(3)
1.5 How closely does this sum approximate the integral?
15(4)
1.6 An arctangent series
19(2)
1.7 Geometric mean rates
21(3)
1.8 Limits of the weighted mean ratio
24(3)
1.9 Limits of mean recurrences
27(5)
1.10 A disguised half-angle iteration
32(4)
1.11 Nested radicals and generalized Fibonacci numbers
36(4)
1.12 A limit involving arctangent
40(3)
1.13 Summing to the double factorials
43(4)
1.14 A fractional part sum with Euler's constant
47(3)
1.15 A Putnam/Monthly limit, problem
50(3)
2 Infinite Series
53(62)
2.1 Wilf wants us thinking rationally
53(4)
2.2 Old wine in a new bottle
57(13)
2.3 Another Euler sum
70(5)
2.4 Reciprocal Catalan number sums
75(4)
2.5 Catalan generating function
79(8)
2.6 A series with log and harmonic numbers
87(4)
2.7 A nonlinear harmonic sum
91(4)
2.8 A series involving Riemann zeta values
95(4)
2.9 Abel theorem continued
99(4)
2.10 A convergence test
103(3)
2.11 A power series with an exponential tail
106(4)
2.12 An infinite matrix product
110(5)
3 Integrations
115(66)
3.1 A Lobachevsky integral
115(4)
3.2 Two log gamma integrals
119(4)
3.3 Short gamma products with simple values
123(4)
3.4 Evaluate an integral by Feynman's way
127(4)
3.5 Three ways to evaluate a log-sine integral
131(6)
3.6 A log-product integral
137(2)
3.7 A Putnam problem beyond
139(4)
3.8 A surface integral with many faces
143(5)
3.9 Evaluate a definite integral by the gamma function
148(3)
3.10 Digamma via a double integral
151(4)
3.11 Another double integral
155(6)
3.12 An integral with log and arctangent
161(5)
3.13 Another integral with log and arctangent
166(10)
3.14 An orthonormal function sequence
176(3)
3.15 An definite integral Quickie
179(2)
4 Inequalities
181(54)
4.1 An inequality from Klamkin
181(6)
4.2 Knuth's exponential inequality
187(6)
4.3 Tight bounds for the normal distribution
193(4)
4.4 An inequality due to Knopp
197(7)
4.5 A discrete inequality by integral
204(3)
4.6 An inequality by power series
207(5)
4.7 An inequality by differential equation
212(5)
4.8 Bounds for a reciprocal of log sum
217(5)
4.9 Log-concavity of a partial binomial sum
222(3)
4.10 A bound of divisor sums related to the Riemann hypothesis
225(10)
5 Monthly Miniatures
235(50)
5.1 Value defined by an integral
235(4)
5.2 Another mean value theorem
239(4)
5.3 The product of derivatives by Darboux's theorem
243(3)
5.4 An integral-derivative inequality
246(4)
5.5 Eigenvalues of a (0, 1)-matrix
250(5)
5.6 A matrix of secants
255(8)
5.7 A gcd-weighted trigonometric sum
263(4)
5.8 A lcm-sum weighted Dirichlet series
267(6)
5.9 An infinite sum-product identity
273(7)
5.10 A polynomial zero identity
280(5)
A List of Problems 285(4)
B Glossary 289(14)
Bibliography 303(8)
Index 311
Hongwei Chen received his PHD from North Carolina State University in 1991. He is currently a professor of mathematics at Christopher Newport University. He has published more than fifty research papers in analysis and partial differential equations. He also authored "Excursions in Classical Analysis" published by the Mathematical Association of America.