Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergraduate Research (Classroom Resource Materials)

Excursions in Classical Analysis will introduce students to advanced problem solving and undergraduate research in two ways: it will provide a tour of classical analysis, showcasing a wide variety of problems that are placed in historical context, and it will help students gain mastery of mathematical discovery and proof. The author presents a variety of solutions for the problems in the book. Some solutions reach back to the work of mathematicians like Leonhard Euler while others connect to other beautiful parts of mathematics. Readers will frequently see problems solved by using an idea that might at first glance, might not even seem to apply to that problem. Other solutions employ a specific technique that can be used to solve many different kinds of problems. Excursions emphasizes the rich and elegant interplay between continuous and discrete mathematics by applying induction, recursion, and combinatorics to traditional problems in classical analysis. The book will be useful in students' preparations for mathematics competitions, in undergraduate reading courses and seminars, and in analysis courses as a supplement. The book is also ideal for self study, since the chapters are independent of one another and may be read in any order. Two classical inequalities A new approach for proving inequalities Means generated by an integral The L'Hôpital monotone rule Trigonometric identities via complex numbers Special numbers On a sum of cosecants The gamma products in simple closed forms On the telescoping sums Summation of subseries in closed form Generating functions for powers of Fibonacci numbers Identities for the Fibonacci powers Bernoulli numbers via determinants On some finite trigonometric power sums Power series of (arcsin x)2 Six ways to sum [zeta] 2 Evaluations of some variant Euler sums Interesting series involving binomial coefficients Parametric differentiation and integration Four ways to evaluate the Poisson integral Some irresistible integrals.