A Radical Approach to Real Analysis

Preface
Contents
1. Crisis in Mathematics: Fourier’s Series
	1.1 Background to the Problem
	1.2 Difficulties with the Solution
2. Infinite Summations
	2.1 The Archimedean Understanding
	2.2 Geometric Series.
	2.3 Calculating π
	2.4 Logarithms and the Harmonic Series
	2.5 Taylor Series
	2.6 Emerging Doubts
3. Differentiability and Continuity
	3.1 Differentiability
	3.2 Cauchy and the Mean Value Theorems
	3.3 Continuity
	3.4 Consequences of Continuity
	3.5 Consequences of the Mean Value Theorem
4. The Convergence of Infinite Series
	4.1 The Basic Tests of Convergence
	4.2 Comparison Tests
	4.3 The Convergence of Power Series
	4.4 The Convergence of Fourier Series
5. Understanding Infinite Series
	5.1 Groupings and Rearrangements
	5.2 Cauchy and Continuity
	5.3 Differentiation and Integration
	5.4 Verifying Uniform Convergence
6. Return to Fourier Series
	6.1 Dirichlet’s Theorem
	6.2 The Cauchy Integral
	6.3 The Riemann Integral
	6.4 Continuity without Differentiability
7. Epilogue
A. Explorations of the Infinite
	A.1 Wallis on π
	A.2 Bernoulli’s Numbers
	A.3 Sums of Negative Powers
	A.4 The Size of n!
B. Bibliography
C. Hints to Selected Exercises
2.1.6
2.4.10
3.1.4
3.3.18
3.4.13
4.1.3
4.3.2
4.4.12
5.4.1
6.1.6
6.3.16
A.1.12
Index
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Corrections
Resources for 
A Radical Approach to Real Analysis (2nd edition)
	Chapter 1: Crises in Mathematics: Fourier\'s Series
		Derivation of Fourier’s Solution
		Laplace’s Equation
		How Fourier found the coeffcients for equation (1.7)
		Approximating Fourier\'s Solution (Maple code)
		The General Solution
		The Orthogonality Relation
		Fourier Series as Complex Power Series
		Maple code for exercises in section 1.2
	Chapter 2: Infinite Summations
		The Quadrature of the Parabolic Segment
		The Archimedean Principle
		Explorations of the Alternating Harmonic Series (Maple code)
		Assigning Values to Divergent Series
		More Pi (Maple code)
		Newton’s Formula
		Explorations of the Harmonic Series
		Euler’s Solution to the Vibrating Drumhead
		Explorations of d\'Alembert\'s Series (Maple code)
		Explorations of Lagrange\'s Remainder (Maple code)
		Maple code for exercises in section 2.1
		Maple code for exercises in section 2.2
		Maple code for exercises in section 2.3
		Maple code for exercises in section 2.4
		Maple code for exercises in section 2.5
	Chapter 3: Differentiability and Continuity
		Newton-Raphson Method
		How to find and write a proof
		Continued Fractions
		The Marquis de l’Hospital
		Maple code for exercise in section 3.3
		Maple code for exercises in section 3.4
		Maple code for exercises in Newton-Raphson Method
	Chapter 4: The Convergence of Infinite Series
		Stirling\'s Formula (Maple code)
		Exponential Function
		Exponential Function (Maple code)
		Convergence in Norm
		Gauss’s Test
		Maple code for exercises in section 4.1
		Maple code for exercises in section 4.2
		Maple code for exercises in section 4.3
		Maple code for exercises in section 4.4
	Chapter 5: Understanding Infinite Series
		The Dilogarithm
		Maple code for exercises in section 5.1
		Maple code for exercises in section 5.2
		Maple code for exercises in section 5.3
	Chapter 6: Return to Fourier Series
		Maple code for exercises in section 6.1
		Maple code for exercises in section 6.2
		Maple code for exercises in section 6.3
		Maple code for exercises in section 6.4
	Appendix A: Explorations of the Infinite
		Binomial Coefficients and Sums of nth Powers
		Maple code for exercises in section A.1
		Maple code for exercises in section A.2
		Maple code for exercises in section A.3
		Maple code for exercises in section A.4
	Acknowledgements
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