**Part I: Some Standard Curriculum**

**Chapter 1: Very Naive Set Theory, Functions, and Proofs**

The Algebra of Sets and the Language of Mathematics

Set Theory and Mathematical Statements

What Are Functions?

**Chapter 2: Numbers, Numbers, and More Numbers**

The Natural Numbers

The Principle of Mathematical Induction

The Integers

Primes and the Fundamental Theorem of Arithmetic

Decimal Representations of Integers

Real Numbers: Rational and “Mostly” Irrational

The Completeness Axiom of $\mathbb R$ and Its Consequences

Construction of the Real Numbers via Dedekind Cuts

$m$-Dimensional Euclidean Space

The Complex Number System

Cardinality and “Most” Real Numbers Are Irrational

**Chapter 3: Infinite Sequences of Real and Complex Numbers**

Convergence and $\epsilon-N$ Arguments for Limits of Sequences

A Potpourri of Limit Properties for Sequences

The Monotone Criteria, the Bolzano–Weierstrass Theorem, and $e$

Completeness, the Cauchy Criterion, and Contractive Sequences

Baby Infinite Series

Absolute Convergence and a Potpourri of Convergence Tests

Tannery’s Theorem and Defining the Exponential Function $\exp(z)$

Decimals and “Most” Real Numbers Are Irrational

**Chapter 4: Limits, Continuity, and Elementary Functions**

Continuity and $\epsilon-\delta$ Arguments for Limits of Functions

A Potpourri of Limit Properties for Functions

Continuity, Thomae’s Function, and Volterra’s Theorem

Compactness, Connectedness, and Continuous Functions

Amazing Consequences of Continuity

Monotone Functions and Their Inverses

Exponentials, Logs, Euler and Mascheroni, and the $\zeta$-Function

Proofs that $\sum1/p$ Diverges

Defining the Trig Functions and $\pi$, and Whichis Larger, $\pi^e$ or $e^\pi$?

Three Proofs of the Fundamental Theorem of Algebra (FTA)

The Inverse Trigonometric Functions and the Complex Logarithm

The Amazing $\pi$ and Its Computation from Ancient Times

**Chapter 5: Some of the Most Beautiful Formulas in the World I–III**

Beautiful Formulas I: Euler, Wallis, and Viète

Beautiful Formuls II: Euler, Gregory, Leibniz, and Madhava

Beautiful Formulas III: Euler’s Formula for $\zeta(2k)$

**Part II: Extracurricular Activities**

**Chapter 6: Advanced Theory of Infinite Series**

Summation by Parts, Bounded Variation, and Alternating Series

Lim Infs/Sups, Ratio/Roots, and Power Series

A Potpourri of Ratio-Type Tests and “Big $\mathcal O$” Notation

Pretty Powerful Properties of Power Series

Cauchy’s Double Series Theorem and A $\zeta$-Function Identity

Rearrangements and Multiplication of Power Series

Composition of Power Series and Bernoulli and Euler Numbers

The Logarithmic, Binomial, Arctangent Series, and $\gamma$

$\pi$, Euler, Fibonacci, Leibniz, Madhava, and Machin

Another Proof that ${\pi^2}/6=\sum_{n=1}^\infty1/{n^2}$ (The Basel Problem)

**Chapter 7: More on the Infinite: Products and Partial Fractions**

Introduction to Infinite Products

Absolute Convergence for Infinite Products

Euler and Tannery: Product Expansions Galore

Partial Fraction Expansions of the Trigonometric Functions

More Proofs that ${\pi^2}/6=\sum_{n=1}^\infty1/{n^2}$

Riemann’s Remarkable $\zeta$-Function, Probability, and ${\pi^2}/6$

Some of the Most Beautiful Formulas in the World IV

**Chapter 8: Infinite Continued Fractions**

Introduction to Continued Fractions

Some of the Most Beautiful Formulas in the World V

Recurrence Relations, Diophantus’s Tomb, and Shipwrecked Sailors

Convergence Theorems for Infinite Continued Fractions

Diophantine Approximations and the Mystery of $\pi$ Solved!

Continued Fractions, Calendars, and Musical Scales

The Elementary Functions and the Irrationality of $e^{p/q}$

Quadratic Irrationals and Periodic Continued Fractions

Archimedes’s Crazy Cattle Conundrum and Diophantine Equations

Epilogue: Transcendental Numbers, $\pi$, $e$, and Where’s Calculus?