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Solutions to Exercises of the book
Amazing and Aesthetic Aspects of Analysis
by Paul Loya
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?
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