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Amazing and Aesthetic Aspects of Analysis: On the incredible infinite (A Course in Undergraduate Analysis, Fall 2006)

Witasek insists, rather, that since aesthetic enjoyment is a genuine pleasure, it must be related to some genuine object of an appropriate sort. But what could this object be, in cases where our aesthetic pleasure is related to Gestalt structures of expression? Note, first of all, that here the genuine feeling of aesthetic pleasure as it unfolds through time manifests a dependence on and a sensitivity to the empathetic-sympathetic emotional arousal with which it is associated. Now the latter is a real phenomenon, which also manifests a real temporal unfolding. Witasek therefore suggests that aesthetic. | Amazing and Aesthetic Aspects of Analysis On the incredible infinite A Course in Undergraduate Analysis Fall 2006 n2 1 1 11 -6 - 12 2 32 42 _ 22 32 52 72 112 - 22 - 1 32 - 1 52 - 1 72 - 1 112 - 1 1 14 12----------------------------2 4----------------------- 12 22----------------------2 4--------------- 22 3 2 - ------- 3 44------ 32 42 ------------------ 42 52 - . Paul Loya This book is free and may not be sold. Please email paul@math.binghamton.edu to report errors or give criticisms Contents Preface i Acknowledgement iii Some of the most beautiful formula in the world v A word to the student vii Part 1. Some standard curriculum 1 Chapter 1. Sets functions and proofs 3 1.1. The algebra of sets and the language of mathematics 4 1.2. Set theory and mathematical statements 11 1.3. What are functions 15 Chapter 2. Numbers numbers and more numbers 21 2.1. The natural numbers 22 2.2. The principle of mathematical induction 27 2.3. The integers 35 2.4. Primes and the fundamental theorem of arithmetic 41 2.5. Decimal representations of integers 49 2.6. Real numbers Rational and mostly irrational 53 2.7. The completeness axiom of R and its consequences 63 2.8. m-dimensional Euclidean space 72 2.9. The complex number system 79 2.10. Cardinality and most real numbers are transcendental 83 Chapter 3. Infinite sequences of real and complex numbers 93 3.1. Convergence and E-N arguments for limits of sequences 94 3.2. A potpourri of limit properties for sequences 102 3.3. The monotone criteria the Bolzano-Weierstrass theorem and e 111 3.4. Completeness and the Cauchy criteria for convergence 117 3.5. Baby infinite series 123 3.6. Absolute convergence and a potpourri of convergence tests 131 3.7. Tannery s theorem the exponential function and the number e 138 3.8. Decimals and most numbers are transcendental á la Cantor 146 Chapter 4. Limits continuity and elementary functions 153 4.1. Convergence and E-Ỗ arguments for limits of functions 154 4.2. A potpourri of limit properties