(Ebook PDF) Real Analysis and Infinity 1st Edition by Hassan Sedaghat ISBN 9780192649539 0192649531 full chapters

• 1 Manifestations of Infinity: An Overview
• 1.1 Infinity within each number
• 1.2 More than one infinity?
• 1.3 Infinite processes
• 1.4 Trigonometric functions and logarithms: Infinity built in
• 1.5 Exercises
• 2 Sets, Functions, Logic, and Countability
• 2.1 Sets and relations
• 2.2 Functions and their basic properties
• 2.3 Basics of logic concepts and operations
• 2.4 Mathematical induction
• 2.5 Bijections and cardinality
• 2.6 An infinity of infinities: Cantor’s theorem
• 2.7 Countable or uncountable?
• 2.8 Exercises
• 3 Sequences and Limits
• 3.1 Infinite lists of numbers
• 3.2 Sequence types and plots
• 3.3 Monotone sequences and oscillating sequences
• 3.4 Convergent sequences and limits
• 3.5 Bounded, unbounded, and divergent sequences
• 3.6 Subsequences
• 3.7 Limit supremum and limit infimum
• 3.8 Sequences, functions, and infinite direct products of sets
• 3.9 Exercises
• 4 The Real Numbers
• 4.1 Rational numbers
• 4.2 Cauchy sequences
• 4.3 Real numbers
• 4.4 Completeness and other foundational theorems of real analysis
• 4.5 The set of real numbers is uncountable
• 4.6 Exercises
• 5 Infinite Series of Constants
• 5.1 On adding infinitely many numbers
• 5.2 Infinite series as limits of sequences of finite sums
• 5.3 The geometric series
• 5.4 Cauchy criterion and convergence tests
• 5.5 Alternating series, conditional and absolute convergence
• 5.6 Real numbers as infinite series, Liouville numbers
• 5.7 Exercises
• 6 Differentiation and Continuity
• 6.1 Velocity, slope, and the derivative
• 6.2 Differentiation rules and higher derivatives
• 6.2.5 When derivatives fail to exist
• 6.3 Continuous functions
• 6.4 Limits via neighborhoods: The ε and δ
• 6.5 The mean value theorem
• 6.6 Indeterminate forms and l’Hospital’s rule
• 6.7 The Newton–Raphson approximation method
• 6.8 Exercises
• 7 Integration
• 7.1 From velocity to position, from curve to area
• 7.2 Riemann integration and integrable functions
• 7.3 Properties of the integral
• 7.4 The Fundamental Theorem of Calculus
• 7.5 Logarithmic and exponential functions
• 7.6 The improper Riemann integral
• 7.7 The integral test for infinite series
• 7.8 Exercises
• 8 Infinite Sequences and Series of Functions
• 8.1 The geometric series as a function series
• 8.2 Convergence of an infinite sequence of functions
• 8.3 Infinite series of functions
• 8.4 Power series expansions of functions
• 8.5 The Taylor series expansion of functions
• 8.6 Trigonometric series and Fourier expansions: A glance
• 8.7 The Weierstrass approximation theorem
• 8.8 Exercises
• 9 Appendices
• 9.1 Appendix: Cantor’s construction, additional detail
• 9.2 Appendix: Discontinuity in a space of functions
• 9.3 Numbered Elements in every chapter

References and Further Reading

Index