A Compact Capstone Course in Classical Calculus

✍ Scribed by Peter R. Mercer

• Publisher: Birkhäuser
• Year: 2023
• Tongue: English
• Leaves: 161
• Series: Compact Textbooks in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

This textbook offers undergraduates a self-contained introduction to advanced topics not covered in a standard calculus sequence. The author’s enthusiastic and engaging style makes this material, which typically requires a substantial amount of study, accessible to students with minimal prerequisites. Readers will gain a broad knowledge of the area, with approaches based on those found in recent literature, as well as historical remarks that deepen the exposition. Specific topics covered include the binomial theorem, the harmonic series, Euler's constant, geometric probability, and much more. Over the fifteen chapters, readers will discover the elegance of calculus and the pivotal role it plays within mathematics.

A Compact Capstone Course in Classical Calculus is ideal for exploring interesting topics in mathematics beyond the standard calculus sequence, particularly for undergraduates who may not be taking more advanced math courses. It would also serve as a useful supplement for a calculus course and a valuable resource for self-study. Readers are expected to have completed two one-semester college calculus courses.

✦ Table of Contents

Preface
Acknowledgments
Contents
1 Prelude: Viète's Product
1.1 Stuff We'll Need
1.2 Proof of Viète's Product
1.3 Seidel's Product
1.4 Epilogue
References
2 Calculus Warm-Up
2.1 Taylor's Theorem
2.2 Integration by Parts
2.3 The Fundamental Property of the Real Numbers
Exercises
References
3 The Probability Integral and Gamma Function
3.1 Evaluating the Probability Integral
3.2 Interpolating the Factorial
3.3 Why This One?
3.4 The Gamma Function
3.5 Epilogue
Exercises
References
4 Wallis's Product
4.1 Prep Work
4.2 Proof of Wallis's Product
4.3 An Example
Exercises
References
5 Interlude: How Big Is a Ball?
5.1 Stuff We'll Need
5.2 Okay, so How Big Is a Ball?
5.3 But How Big Is a Ball, in a Really Big Place?
References
6 Convexity—Tangents
6.1 Convexity and Tangent Lines
6.2 Wallis's Product and Probability Integral, Again
Exercises
References
7 Some Important Series
7.1 p-Series
7.2 Alternating Harmonic Series and Madhavan Series, Etc.
7.3 Euler's Constant
Exercises
References
8 Geometric Probability
8.1 Enter: Calculus
8.2 Buffon's Needle
8.3 Epilogue
Exercises
References
9 Convexity—Chords
9.1 Convexity and Chords
9.2 Jensen's Inequality
9.3 Means Means Means
Exercises
References
10 Interlude: Minkowski Distance
10.1 Convex Sets
10.2 The Triangle Inequality
10.3 From One Extreme to the Other
10.4 Epilogue
References
11 The Basel Problem
11.1 The Basel Problem—Solution
11.2 A Cool Example
11.3 The Sum of the Reciprocals of the Primes
Exercises
References
12 Interlude: Beyond Basel
12.1 The Bernoulli Polynomials
12.2 Toward ζ(2m)
12.3 The Formula for ζ(2m)
12.4 Sums of Powers
12.5 Epilogue
References
13 Stirling's Formula
13.1 Prep Work
13.2 de Moivre's Version of Stirling's Formula
Exercises
References
14 Euler's Sine Product
14.1 The Sine Product
14.2 Amazing Consequences
Exercises
References
15 Postlude: Stirling's Formula Again
15.1 Prep Work
15.2 Stirling's Version of Stirling's Formula
15.3 Epilogue
References
Index

✦ Subjects

Classical calculus; Binomial Theorem; e Irrationality; Harmonic Series; Madhaven Series; Seidel's Product; Kepler-Bowkamp Constant; Euler's Constant; Apery's Constant; Bernoulli's Inequality; Geometric Probability; Minkowski Distance; Basel Problem; Stirling's Formula; Euler's Sine Product