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AMS / MAA CLASSROOM RESOURCE MATERIALS VOL 35 I I “Master” — 2011/5/9 — 10:51 — Page I — #1 I I AMS / MAA CLASSROOM RESOURCE MATERIALS VOL 35 i i “master” — 2011/5/9 — 10:51 — page i — #1 i i 10.1090/clrm/035 Excursions in Classical Analysis Pathways to Advanced Problem Solving and Undergraduate Research i i i i i i “master” — 2011/5/9 — 10:51 — page ii — #2 i i Chapter 11 (Generating Functions for Powers of Fibonacci Numbers) is a reworked version of material from my same name paper in the International Journalof Mathematical Education in Science and Tech- nology, Vol. 38:4 (2007) pp. 531–537. www.informaworld.com Chapter 12 (Identities for the Fibonacci Powers) is a reworked version of material from my same name paper in the International Journal of Mathematical Education in Science and Technology, Vol. 39:4 (2008) pp. 534–541. www.informaworld.com Chapter 13 (Bernoulli Numbers via Determinants)is a reworked ver- sion of material from my same name paper in the International Jour- nal of Mathematical Education in Science and Technology, Vol. 34:2 (2003) pp. 291–297. www.informaworld.com Chapter 19 (Parametric Differentiation and Integration) is a reworked version of material from my same name paper in the Interna- tional Journal of Mathematical Education in Science and Technology, Vol. 40:4 (2009) pp. 559–570. www.informaworld.com c 2010 by the Mathematical Association of America, Inc. Library of Congress Catalog Card Number 2010924991 Print ISBN 978-0-88385-768-7 Electronic ISBN 978-0-88385-935-3 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “master” — 2011/5/9 — 10:51 — page iii — #3 i i Excursions in Classical Analysis Pathways to Advanced Problem Solving and Undergraduate Research Hongwei Chen Christopher Newport University Published and Distributed by The Mathematical Association of America i i i i i i “master” — 2011/5/9 — 10:51 — page iv — #4 i i Committee on Books Frank Farris, Chair Classroom Resource Materials Editorial Board Gerald M. Bryce, Editor Michael Bardzell William C. Bauldry Diane L. Herrmann Wayne Roberts Susan G. Staples Philip D. Straffin Holly S. Zullo i i i i i i “master” — 2011/5/9 — 10:51 — page v — #5 i i CLASSROOM RESOURCE MATERIALS Classroom Resource Materials is intended to provide supplementary classroom material for students—laboratory exercises, projects, historical information, textbooks with unusual approaches for presenting mathematical ideas, career information, etc. 101 Careers in Mathematics, 2nd edition edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein The Calculus Collection: A Resource for AP and Beyond, edited by Caren L. Diefenderfer and Roger B. Nelsen Calculus Mysteries and Thrillers, R. Grant Woods Conjecture and Proof, Mikl´os Laczkovich Counterexamples in Calculus, Sergiy Klymchuk Creative Mathematics, H. S. Wall Environmental Mathematics in the Classroom, edited by B. A. Fusaro and P. C. Kenschaft Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergrad- uate Research, by Hongwei Chen Exploratory Examples for Real Analysis, Joanne E. Snow and Kirk E. Weller Geometry From Africa: Mathematical and Educational Explorations, Paulus Gerdes Historical Modules for the Teaching and Learning of Mathematics (CD), edited by Victor Katz and Karen Dee Michalowicz Identification Numbers and Check Digit Schemes, Joseph Kirtland Interdisciplinary Lively Application Projects, edited by Chris Arney Inverse Problems: Activities for Undergraduates, Charles W. Groetsch Laboratory Experiences in Group Theory, Ellen Maycock Parker Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and Victor Katz Math Made Visual: Creating Images for Understanding Mathematics, Claudi Alsina and Roger B. Nelsen Ordinary Differential Equations: A Brief Eclectic Tour, David A. S´anchez Oval Track and Other Permutation Puzzles, John O. Kiltinen A Primer of Abstract Mathematics, Robert B. Ash Proofs Without Words, Roger B. Nelsen Proofs Without Words II, Roger B. Nelsen She Does Math!, edited by Marla Parker Solve This: Math Activities for Students and Clubs, James S. Tanton Student Manual for Mathematics for Business Decisions Part 1: Probability and Simula- tion, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic i i i i i i “master” — 2011/5/9 — 10:51 — page vi — #6 i i Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimiza- tion, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Teaching Statistics Using Baseball, Jim Albert Visual Group Theory, Nathan C. Carter Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, Annalisa Crannell, Gavin LaRose, Thomas Ratliff, Elyn Rykken MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 i i i i i i “master” — 2011/5/9 — 10:51 — page vii — #7 i i Contents Preface xi 1 Two Classical Inequalities 1 1.1 AM-GMInequality ............................. 1 1.2 Cauchy-SchwarzInequality . 8 Exercises ..................................... 12 References..................................... 15 2 A New Approach for Proving Inequalities 17 Exercises ..................................... 22 References..................................... 23 3 Means Generated by an Integral 25 Exercises ..................................... 30 References..................................... 32 4 The L’Hˆopital Monotone Rule 33 Exercises ..................................... 37 References..................................... 38 5 Trigonometric Identities via Complex Numbers 39 5.1 APrimerofcomplexnumbers . 39 5.2 FiniteProductIdentities . .. 41 5.3 FiniteSummationIdentities . .. 43 5.4 Euler’sInfiniteProduct . 45 5.5 Sumsofinversetangents ..... ........ ....... ...... 48 5.6 TwoApplications .............................. 49 Exercises ..................................... 51 References..................................... 53 6 Special Numbers 55 6.1 GeneratingFunctions ....... ........ ....... ...... 55 6.2 FibonacciNumbers ............................. 56 6.3 Harmonicnumbers ............................. 58 6.4 BernoulliNumbers ............................. 61 vii i i i i i i “master” — 2011/5/9 — 10:51 — page viii — #8 i i viii Contents Exercises ..................................... 69 References..................................... 72 7 On a Sum of Cosecants 73 7.1 Awell-knownsumanditsgeneralization . .... 73 7.2 Roughestimates............................... 74 7.3 Tyinguptheloosebounds..... ........ ....... ...... 76 7.4 FinalRemarks................................ 79 Exercises ..................................... 79 References..................................... 81 8 The Gamma Products in Simple Closed Forms 83 Exercises ..................................... 89 References..................................... 91 9 On the Telescoping Sums 93 9.1 Thesumofproductsofarithmeticsequences . ..... 94 9.2 The sum of products of reciprocals of arithmetic sequences ........ 95 9.3 Trigonometricsums ............................. 97 9.4 Somemoretelescopingsums . 101 Exercises ..................................... 106 References..................................... 108 10 Summation of Subseries in Closed Form 109 Exercises ..................................... 117 References..................................... 119 11 Generating Functions for Powers of Fibonacci Numbers 121 Exercises ..................................... 128 References..................................... 130 12 Identities for the Fibonacci Powers 131 Exercises ..................................... 140 References..................................... 142 13 Bernoulli Numbers via Determinants 143 Exercises ..................................... 149 References..................................... 152 14 On Some Finite Trigonometric Power Sums 153 14.1 Sums involving sec2p .k=n/ ........................ 154 14.2 Sums involving csc2p .k=n/ ........................ 156 14.3 Sums involving tan2p .k=n/ ........................ 157 14.4 Sums involving cot2p .k=n/ ........................ 159 Exercises ..................................... 162 References..................................... 164 15 Power Series of .arcsin x/2 165 15.1FirstProofoftheSeries(15.1) . ... 165 15.2SecondProofoftheSeries(15.1) . ... 167 Exercises ..................................... 171 i i i i i i “master” — 2011/5/9 — 10:51 — page ix — #9 i i Contents ix References..................................... 173 16 Six Ways to Sum .2/ 175 16.1Euler’sProof................................. 176 16.2ProofbyDoubleIntegrals . 177 16.3ProofbyTrigonometricIdentities . ..... 181 16.4ProofbyPowerSeries....... ........ ....... ...... 182 16.5ProofbyFourierSeries . 183 16.6ProofbyComplexVariables. 183 Exercises ..................................... 184 References..................................... 187 17EvaluationsofSomeVariantEulerSums 189 Exercises ..................................... 197 References..................................... 199 18 Interesting Series Involving Binomial Coefficients 201 18.1 An integralrepresentation and its applications . .......... 202 18.2SomeExtensions .............................. 208 18.3 Searching for new formulas for ...................... 210 Exercises ..................................... 212 References..................................... 215 19ParametricDifferentiationandIntegration 217 Example1..................................... 217 Example2..................................... 218 Example3..................................... 219 Example4..................................... 220
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