A Concrete Introduction to Real Analysis

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Overview

Most volumes in analysis plunge students into a challenging new mathematical environment, replete with axioms, powerful abstractions, and an overriding emphasis on formal proofs. This can lead even students with a solid mathematical aptitude to often feel bewildered and discouraged by the theoretical treatment. Avoiding unnecessary abstractions to provide an accessible presentation of the material, A Concrete Introduction to Real Analysis supplies the crucial transition from a calculations-focused treatment of mathematics to a proof-centered approach.

Drawing from the history of mathematics and practical applications, this volume uses problems emerging from calculus to introduce themes of estimation, approximation, and convergence. The book covers discrete calculus, selected area computations, Taylor's theorem, infinite sequences and series, limits, continuity and differentiability of functions, the Riemann integral, and much more. It contains a large collection of examples and exercises, ranging from simple problems that allow students to check their understanding of the concepts to challenging problems that develop new material.

Providing a solid foundation in analysis, A Concrete Introduction to Real Analysis demonstrates that the mathematical treatments described in the text will be valuable both for students planning to study more analysis and for those who are less inclined to take another analysis class.

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Product Details

Table of Contents

DISCRETE CALCULUS
Introduction
Proof by Induction
A Calculus of Sums and Differences
Sums of Powers
Problems
SELECTED AREA COMPUTATIONS
Introduction
Areas under Power Function Graphs
The Computation of p
Natural Logarithms
Stirling's Formula
Problems
LIMITS AND TAYLOR'S THEOREM
Introduction
Limits of Infinite Sequences
Series Representations
Taylor Series
Problems
INFINITE SERIES
Introduction
Positive Series
General Series
Grouping and Rearrangement
Problems
A BIT OF LOGIC
Somemathematical Philosophy
Propositional Logic
Predicates and Quantifiers
Proofs
Problems
REAL NUMBERS
Field Axioms
Order Axioms
Completeness Axioms
Subsequences and Compact Intervals
Products and Fractions
Problems
FUNCTIONS
Introduction
Basics
Limits and Continuity
Derivatives
Problems
INTEGRALS
Introduction
Integrable Functions
Properties of Integrals
Numerical Computation of Integrals
Problems
MORE INTEGRALS
Introduction
Improper Integrals
Integrals with Parameters
Problems
REFERENCES
INDEX

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