Amazing and aesthetic aspects of analysis

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Amazing and Aesthetic Aspects of Analysis: On the incredible infinite (A Course in Undergraduate Analysis, Fall 2006) π2 1 1 = + + + + ··· 22 32 52 72 112 · · · · ··· = 2 − 32 − 52 − 72 − 112 − 1 = 02 + 12 − 14 12 + 22 − 24 22 + 32 − 34 32 + 42 − 44 42 + 52 − Paul Loya (This book is free and may not be sold Please email paul@math.binghamton.edu to report errors or give criticisms) Contents Preface i Acknowledgement iii Some of the most beautiful formulæ in the world A word to the student Part v vii Some standard curriculum Chapter Sets, functions, and proofs 1.1 The algebra of sets and the language of mathematics 1.2 Set theory and mathematical statements 1.3 What are functions? 11 15 Chapter Numbers, numbers, and more numbers 2.1 The natural numbers 2.2 The principle of mathematical induction 2.3 The integers 2.4 Primes and the fundamental theorem of arithmetic 2.5 Decimal representations of integers 2.6 Real numbers: Rational and “mostly” irrational 2.7 The completeness axiom of R and its consequences 2.8 m-dimensional Euclidean space 2.9 The complex number system 2.10 Cardinality and “most” real numbers are transcendental 21 22 27 35 41 49 53 63 72 79 83 Chapter Infinite sequences of real and complex numbers 3.1 Convergence and ε-N arguments for limits of sequences 3.2 A potpourri of limit properties for sequences 3.3 The monotone criteria, the Bolzano-Weierstrass theorem, and e 3.4 Completeness and the Cauchy criteria for convergence 3.5 Baby infinite series 3.6 Absolute convergence and a potpourri of convergence tests 3.7 Tannery’s theorem, the exponential function, and the number e 3.8 Decimals and “most” numbers are transcendental ´a la Cantor 93 94 102 111 117 123 131 138 146 Chapter Limits, continuity, and elementary functions 4.1 Convergence and ε-δ arguments for limits of functions 4.2 A potpourri of limit properties for functions 4.3 Continuity, Thomae’s function, and Volterra’s theorem 153 154 160 166 iii iv CONTENTS 4.4 4.5 4.6 4.7 4.8 4.9 4.10 Compactness, connectedness, and continuous functions Monotone functions and their inverses Exponentials, logs, Euler and Mascheroni, and the ζ-function The trig functions, the number π, and which is larger, π e or eπ ? Three proofs of the fundamental theorem of algebra (FTA) The inverse trigonometric functions and the complex logarithm The amazing π and its computations from ancient times 172 182 187 198 210 217 226 Chapter Some of the most beautiful formulæ in the world 5.1 Euler, Wallis, and Vi`ete 5.2 Euler, Gregory, Leibniz, and Madhava 5.3 Euler’s formula for ζ(2k) 237 240 249 258 Part 267 Extracurricular activities Chapter Advanced theory of infinite series 6.1 Summation by parts, bounded variation, and alternating series 6.2 Liminfs/sups, ratio/roots, and power series 6.3 A potpourri of ratio-type tests and “big O” notation 6.4 Some pretty powerful properties of power series 6.5 Double sequences, double series, and a ζ-function identity 6.6 Rearrangements and multiplication of power series 6.7 Proofs that 1/p diverges 6.8 Composition of power series and Bernoulli and Euler numbers 6.9 The logarithmic, binomial, arctangent series, and γ 6.10 π, Euler, Fibonacci, Leibniz, Madhava, and Machin ∞ 6.11 Another proof that π /6 = n=1 1/n2 (The Basel problem) 269 271 279 290 295 300 311 320 325 332 340 344 Chapter More on the infinite: Products and partial fractions 7.1 Introduction to infinite products 7.2 Absolute convergence for infinite products 7.3 Euler, Tannery, and Wallis: Product expansions galore 7.4 Partial fraction expansions of the trigonometric functions ∞ 7.5 More proofs that π /6 = n=1 1/n2 7.6 Riemann’s remarkable ζ-function, probability, and π /6 7.7 Some of the most beautiful formulæ in the world IV 349 350 355 359 366 370 373 382 Chapter Infinite continued fractions 8.1 Introduction to continued fractions 8.2 Some of the most beautiful formulæ in the world V 8.3 Recurrence relations, Diophantus’ tomb, and shipwrecked sailors 8.4 Convergence theorems for infinite continued fractions 8.5 Diophantine approximations and the mystery of π solved! 8.6 Continued fractions and calendars, and math and music 8.7 The elementary functions and the irrationality of ep/q 8.8 Quadratic irrationals and periodic continued fractions 8.9 Archimedes’ crazy cattle conundrum and diophantine equations 8.10 Epilogue: Transcendental numbers, π, e, and where’s calculus? 389 390 394 403 411 422 433 437 446 456 464 Bibliography 471 CONTENTS Index v 481 Preface I have truly enjoyed writing this book Admittedly, some of the writing is too overdone (e.g overdoing alliteration at times), but what can I say, I was having fun The “starred” sections of the book are meant to be “just for fun” and don’t interfere with other sections (besides perhaps other starred sections) Most of the quotes that you’ll find in these pages are taken from the website http://www-gap.dcs.st-and.ac.uk/~history/Quotations/ This is a first draft, so please email me any errors, suggestions, comments etc about the book to paul@math.binghamton.edu The overarching goals of this textbook are similar to any advanced math textbook, regardless of the subject: Goals of this textbook The student will be able to • comprehend and write mathematical reasonings and proofs • wield the language of mathematics in a precise and effective manner • state the fundamental ideas, axioms, definitions, and theorems upon which real analysis is built and flourishes • articulate the need for abstraction and the development of mathematical tools and techniques in a general setting The objectives of this book make up the framework of how these goals will be accomplished, and more or less follow the chapter headings: Objectives of this textbook The student will be able to • identify the interconnections between set theory and mathematical statements and proofs • state the fundamental axioms of the natural, integer, and real number systems and how the completeness axiom of the real number system distinguishes this system from the rational system in a powerful way • apply the rigorous ε-N definition of convergence for sequences and series and recognize monotone and Cauchy sequences • apply the rigorous ε-δ definition of limits for functions and continuity and the fundamental theorems of continuous functions • determine the convergence and properties of an infinite series, product, or continued fraction using various tests • identify series, product, and continued fraction formulæ for the various elementary functions and constants I’d like to thank Brett Bernstein for looking over the notes and gave many valuable suggestions Finally, some last words about my book This not a history book (but we try to talk history throughout this book) and this not a “little” book like Herbert Westren Turnbull’s book The Great Mathematicians, but like Turnbull, I hope i ii PREFACE If this little book perhaps may bring to some, whose acquaintance with mathematics is full of toil and drudgery, a knowledge of those great spirits who have found in it an inspiration and delight, the story has not been told in vain There is a largeness about mathematics that transcends race and time: mathematics may humbly help in the market-place, but it also reaches to the stars To one, mathematics is a game (but what a game!) and to another it is the handmaiden of theology The greatest mathematics has the simplicity and inevitableness of supreme poetry and music, standing on the borderland of all that is wonderful in Science, and all that is beautiful in Art Mathematics transfigures the fortuitous concourse of atoms into the tracery of the finger of God Herbert Westren Turnbull (1885–1961) Quoted from [225, p 141] Paul Loya Binghamton University, Vestal Parkway, Binghamton, NY 13902 paul@math.binghamton.edu Soli Deo Gloria Acknowledgement To Jesus, my Lord, Savior and Friend iii Index series error estimate, 275 series test, 275 Angle, 78, 206 Anthoniszoon, Adriaan, 228, 423 Ap´ ery, Roger, 263 Approximable numbers, 464 Approximation(s) best, 424 good, 423 1/2 √ π ≈ 10, 227 , 311/3 , 4930 23 159 432 π ≈ 56 Φ2 , 414 Abel summable, 299 Abel’s lemma, 271 Abel’s limit theorem, 296 Abel’s multiplication theorem, 317 Abel’s test for series, 277 Abel, Neils, 296 Abel, Niels, 123, 271 Absolute convergence double series, 302 infinite products, 355 series, 135 Absolute convergence theorem, 135 Absolute value, 17, 39 of complex numbers, 82 Absolute value rules, 39 Additive function, 171 Additive identity existence for complex numbers, 80 existence for integers, 35 existence for reals, 54 existence for vectors, 73 Additive inverse existence for complex numbers, 80 existence for integers, 35 existence for reals, 54 existence for vectors, 73 AGM method (to compute π), 233 Ahmes, 227 Algebra of limits functions, 161 sequences, 107 Algebraic number, 88 Algorithm Archimedes’, 232 Borchardt’s, 235 Brahmagupta’s, 461 canonical continued fraction, 415 division, 42 Euclidean, 46 Almost integer, 206 Alternating harmonic series, 136, 154, 275 harmonic series rearrangement, 311 log formula, 194 1/4 1/3 , 3061/5 , π ≈ 2143 , 432 22 Approximations eπ√− π ≈ 20, 206 eπ 163 = 262537412640768744, 206 Archimedean ordering of the natural numbers, 25 property of reals, 68 Archimedes of Syracuse, 153, 226, 227, 423, 456 Archimedes’ algorithm, 232 Archimedes’ cattle problem, 456 Archimedes’ three propositions, 228 Argument of a complex number, 219 principal, 219 Ariel, 437 Aristotle, 182 Aristoxenus, 437 Arithmetic mean, 34 Arithmetic properties of series theorem, 127 Arithmetic-geometric mean inequality, 34, 195 application to e, 116 Associative law addition for complex numbers, 80 addition for integers, 35 addition for natural numbers, 22 addition for reals, 53 addition for vectors, 73 multiplication for complex numbers, 80 481 77729 254 1/5 , 482 multiplication for integers, 36 multiplication for natural numbers, 22 multiplication for reals, 54 multiplication for vectors, 73 Axiom(s) completeness, 63 integers, 35 natural numbers, 22 b-adic expansion, see also b-adic expansion b-adic representation of integers, 51 of real numbers, 147 Bachmann, Paul, 291 Ball closed, 77 open, 77 Ball norm, 76 Barber puzzle, Base of a power, 28, 190, 222 to represent integers, 51 to represent real numbers, 146 Basel problem, 250 Bell, Jordan, 238 Bellman, Richard, 321 Bernoulli numbers, 262, 270, 327 Bernoulli’s inequality, 30 Bernoulli, Jacob, 30, 125, 249, 269 numbers, 270, 327 Bernoulli, Nicolaus, 344 Bertrand, Joseph, 293 Best approximation, 424 Best approximation theorem, 425, 429 big O notation, 291 Bijection, 18 Binary system, 50 Binomial coefficient, 31, 332 series, 333, 336 theorem, 31 Bisection method, 181 Bolzano, Bernard, 114 Bolzano-Weierstrass theorem for R, 114 Bolzano-Weierstrass theorem for Rm , 114 Bombelli, Rafael, 414 Borchardt’s algorithm, 235 Borchardt, Carl, 235 Borwein, Jonathan, 233 Borwein, Peter, 233 Bosanquet, Robert, 437 Bounded above, 63 below, 64 sequence, 104 variation, 272 Boundedness theorem, 175 Box norm, 79 INDEX Brahmagupta, 459 quote, 459 zero and negative numbers, 21 Brahmagupta’s algorithm, 461 Brent, Richard, 233 Bromwich, Thomas, 257 Brouncker, Lord William, 233, 389 Brouwer’s fixed point theorem, 179 Brouwer, L E J, 179 Bunyakovski˘ı, Viktor, 74 Caesar, Julius, 433 Calendar Gregorian, 433 Julian, 433 Persian, 437 Canonical (simple) continued fraction, 415 Canonical continued fraction algorithm, 415 Cantor’s diagonal argument, 151 original version, 152 Cantor’s first proof (of the uncountability of R), 86 Cantor’s second proof (of the uncountability of R), 151 Cantor’s theorem, 90 Cantor, Georg, 83 uncountability of transcendental numbers, 89 Cardinality, 83 Cartesian product, 15 Cassini’s identity, 411 Castellanos, Dario, 342 Cataldi, Pietro, 421 Cattle problem, 456 Cauchy condensation test, 133 Cauchy criterion theorem, 119 Cauchy product, 315 Cauchy sequence, 117 Cauchy’s arithmetic mean theorem, 344 Cauchy’s criterion theorem for series, 132 Cauchy’s double series theorem, 145, 305 Cauchy’s multiplication theorem, 318 Cauchy’s root test, 284 Cauchy, Augustin, 74, 315 Cauchy-Bunyakovski˘ı-Schwarz inequality, 74 Cauchy-Hadamard theorem, 287 Cauchy-Schwarz inequality, 74 Chain of intervals, 181 Change of base formula, 196 Characteristic function, 20 Chongzhi, Zu, see also Chung-Chi, Tsu Chung-Chi, Tsu, 227, 423 Cloitre, Benoit, 386 Closed ball, 77 interval, natural number operations, 22 Coconut puzzles, 409 INDEX Codomain of function, 16 Coin game, 34 Commutative law addition for complex numbers, 80 addition for integers, 35 addition for natural numbers, 22 addition for reals, 53 addition for vectors, 73 multiplication for complex numbers, 80 multiplication for integers, 36 multiplication for natural numbers, 22 multiplication for reals, 54 Commutative ring, 36 Compact, 173 Compactness lemma, 173 Comparison test, 132 Complement of sets, Complete quotients, 416 Completeness axiom, 63 Completeness property of R, 64 Completeness property of Rm , 120 Complex logarithm, 221 power, 190, 222 Complex conjugate, 82 Complex numbers (C), 79 Component functions, 161 Component theorem for continuity, 168 for functions, 161 for sequences, 103 Composite, 44 Composition of functions, 17 Composition of limits theorem, 162 confluent hypergeometric limit function, 438 Congruent modulo √ n, 47 √ Conjugate in Z[ d] or Q[ d], 449 Connected set, 173 Connectedness lemma, 174 Constant function, 20 Constant(s) Euler-Mascheroni γ, 153, 193 Continued fraction, 123 canonical, 415 unary, 422 Continued fraction convergence theorem, 416 Continued fraction(s) finite, 391, 393 infinite, 393 nonnegative, 404 regular, 411 simple, 392 terminating, 393 Continued fractions transformation rules of, 394 Continuity at a point, 166 on a set, 167 483 Continuity theorem for power series, 295 Continuous function(s) algebra of, 168 composition of, 168 Continuous functions, 166 Contractive sequence, 120 Contractive sequence theorem, 120 Contrapositive, 6, 11, 24 Convergence double sequences, 300 double series, 302 functions, 154 infinite products, 351 infinite series, 124 sequences, 95 Convergent of a continued fraction, 393 Converse, 13 Convex set, 72 Coprime numbers, 47, 153, 234, 378 Cosecant function, 200 power series, 331 Cosine function, 144, 198 Cotangent continued fraction, 445 Cotangent function, 200 power series of, 329 Countability of algebraic numbers, 89 of rational numbers, 86 Countable set, 83 Countably infinite, 83 Cover of sets, 172 Cube root, 57 d’Alembert’s ratio test, 284 d’Alembert, Jean Le Rond, 284 quote, 95 Davis’ Broadway cafe, 419 de Lagny, Thomas, 343 de Moivre’s formula, 198 de Moivre, Abraham, 198 De Morgan and Bertrand’s test, 293 De Morgan, Augustus, 293 laws, quote, 35 Decimal system, 49 Decimal(s) integers, 49 real numbers, 146 Degree, 202 Dense, 170 Density of the (ir)rationals, 69 Difference of sets, Difference sequence, 122 Digit(s), 49, 147 Diophantine equations, 407 Diophantus of Alexandrea, 389, 407 484 Dirac, Paul, 237 Direct proof, 24 Dirichlet eta function, 376 Dirichlet function, 17, 159 modified, 169 Dirichlet’s approximation theorem, 430 Dirichlet’s test, 273 Dirichlet’s theorem for rearrangements, 314 Dirichlet, Johann, 17, 159, 273 Disconnected set, 173 Discontinuity jump, 183 Disjoint sets, Distance between vectors or points, 76 Distributive law complex numbers, 80 integers, 36 natural numbers, 22 reals, 54 vectors, 73 Divergence infinite products, 351 infinite series, 124 proper (for functions), 164 proper (for sequences), 109 sequences, 95 to ±∞ for functions, 164 to ±∞ for sequences, 109 to zero for infinite products, 351 Divide, divisible, 41 Divisibility rules, 42 Divisibility tricks, 53 Division algorithm, 42 Divisor, 41 Domain of function, 16 dot product, 73 Double a cube, 228 Double sequence, 145, 300 Duodecimal system, 52 Dux, Erich, 321 Dyadic system, 50 e, 93, 114 approximation, 143 infinite nested product formula, 143 irrationality, 143, 276, 420 nonsimple continued fraction, 389, 402 simple continued fraction, 389, 419, 443 e and π in a mirror, 386 Eddington, Sir Arthur, 21 Einstein, Albert, 376 Empty set, Enharmonic Harmonium, 437 ε/2-trick, 102, 119 ε-δ arguments, 156 ε-δ definition of limit, 154 ε-N arguments, 96 ε-N definition of limit, 94 INDEX ε-principle, 102 Equality of functions, 20 of sets, Eratosthenes of Cyrene, 456 Erdă os, Paul, 325 Euclid’s theorem, 44 Euclidean algorithm, 46 Euclidean space, 72 Eudoxus of Cnidus, 68 Euler numbers, 270, 330 Euler’s identity, 198 Euler’s sum for π /6, 126 Euler, Leonhard, 126, 153, 234, 249, 339, 341, 389, 399, 441, 462, 464 e, 114 f (x) notation, 16 identity for eiz , 198 letter to a princess, 435 on the series − + · · · , 93 on transcendental numbers, 88 popularization of π, 228 quote, 41, 93 role played in FTA, 210 summation notation Σ, 28 Euler-Mascheroni constant, 153 Euler-Mascheroni constant γ, 193 Even function, 299 number, 44 Existence of complex n-th roots, 215 of real n-th roots, 65 Exponent, 28, 190, 222 Exponential function, 94, 140, 187, 285 application of Cauchy multiplication, 319 the most important function, 153 Exponential funtion continued fraction, 442 Extended real numbers, 109, 280 Factor, 41 Factorial(s), 31, 46 how to factor, 48 Family of sets, Fandreyer, Ernest, 210 Ferguson, D., 234 Fermat’s last theorem, 462 Fermat’s theorem, 47 Fermat, Pierre, 462 Fermat, Pierre de, 47 Fibonacci sequence, 34, 131, 278, 289, 341, 411 Fibonacci, Leonardo, 341, 418 Field, 54 ordered, 54 Finite, 83 subcover, 172 INDEX subcover property, 173 Flajolet, Phillip, 339, 341 FOIL law, 24 Formula(s) 1/3 e1 · e1/4 · · · , 194 e1/2 e π = ∞ n=0 arctan F2n+1 , 342 n (−1) ∞ 3z 5z n=0 (2n+1)z = 3z +1 · 5z −1 · · · , Euler’s infinite product (sin πx), 238 Euler’s infinite product (sin πz), 349, 359 Euler’s infinite product (sin πz), Proof I, 241 Euler’s infinite product (sin πz), Proof II, 249 Euler’s infinite product (sin πz), Proof III, 361 Euler’s infinite product (sin πz), Proof IV, 362 Euler’s partial fraction ( cosπ πz ), 369 2=2= (−1)n ∞ n=0 (2n+1)2k+1 = E2k (−1)k 2(2k)! 386 π 2k+1 , 385 e = + 22 + 33 + 44 , 402 addition for (co)sine, 199 arctan x = x2 x 32 x + 3−x2 + 5−3x2 + Euler’s partial fraction ( sinππz ), 366, 369 ), 369 Euler’s partial fraction (π tan πz Euler’s partial fraction (πz cot πz), 366 Euler’s partial fraction (π/ sin πz), 350 Euler’s product ( π2 = 32 · 65 · 76 · · · ), 350, 386 Euler’s product ( π4 = 34 · 45 · 78 · · · ), 350, 386 γ =1− ∞ n=2 n ζ(n) − , 339 , 398 Castellano’s, 342 change of base, 196 continued fraction for ex , 442 cosecant power series, 331 cotangent continued fraction, 445 cotangent power series, 329 de Moivre’s, 198 double angle for (co)sine, 199 e nonsimple continued fraction, 389 e simple continued fraction, 389, 443 n e n = limn→∞ n + ··· + n , 191 e−1 n e = limn→∞ 345 e = 21 · π −π e −e 2π 2/x e +1 e2/x −1 · = 16 15 · · · , 352 ∞ n=1 =x+ 1+ 1 3x + 5x n2 485 ··· n+1 n n , 441 Euler’s (π /6), 154, 234, 269 Euler’s (π /6), Proof I, 249 Euler’s (π /6), Proof II, 251 Euler’s (π /6), Proof III, 253 Euler’s (π /6), Proof IV, 257 Euler’s (π /6), Proof V, 258 Euler’s (π /6), Proof VI, 264 Euler’s (π /6), Proof VII, 345 Euler’s (π /6), Proof VIII, 370 Euler’s (π /6), Proof VIIII, 371 Euler’s (π /6), Proof X, 372 Euler’s (π /6), Proof VIIIIII, 383 Euler’s (π /90), 262, 371, 383 Euler’s (π /945), 262, 371, 383 Euler’s formula (η(2k)), 383 Euler’s formula (ζ(2k)), 263, 383 Euler’s infinite product (cos πz), 364 Euler’s infinite product (cos πz), Proof I, 364 Euler’s infinite product (cos πz), Proofs II,III,IV, 364 Euler’s infinite product (π /15), 384 Euler’s infinite product (π /6), 350, 375, 383 Euler’s infinite product (π /90), 384 n−1) ζ(n)− (−1)n 1/n , 362 (−1)n ∞ n=2 n γ = 32 −log 2− , 339 , γ= ∞ ζ(n), 269, 339 n=2 n 4/π continued fraction, 389 Gregory-Leibniz-Madhava, 239 Gregory-Leibniz-Madhava’s, 154, 234, 269 Gregory-Leibniz-Madhava, Proof I, 254 Gregory-Leibniz-Madhava, Proof II, 340 Gregory-Leibniz-Madhava, Proof III, 386 Gregory-Madhava’s arctangent, 338 half-angle for (co)sine, 200 hyperbolic cotangent continued fraction, 439 hyperbolic secant power series, 330 hyperbolic tangent continued fraction, 441 12 x 22 x log(1 + x) = x1 + (2−1x) + (3−2x) , 402 log 2 22 log = 11 11 , 396 + + 1+ log log = ∞ n=2 2n ζ(n), 309 Lord Brouncker’s, 233, 389, 397 Machin’s, 234, 270, 342 partial fraction of 1/ sin2 x, 257 partial fraction of 1/ sin2 x, Proof I, 257 partial fraction of 1/ sin2 x, Proof II, 257 Fn+1 , 422 Φ = limn→∞ F n n−1 (−1) ∞ n=1 Fn Fn+1 , 422 (−1)n Φ−1 = ∞ n=2 Fn Fn+2 , 422 2 π = + 16 + 36 + 56 , 400 (1−x)2 x2 π cot πx = x1 + 1−2x + 2x , 403 π 2·3 = + 11 + 1·2 , 399 + cos πx (x−1)2 (x+1)2 = x + + −2·1 + −2 , π Φ= sin πx πx =1− (1−x)2 (1+x)2 x 1+ 2x + 1−2x 403 , 403 486 sin πx πx INDEX 1·(1−x) 1·(1+x) x 1+ x + 1−x , (1−x) x + 1−2x + 2x , 403 =1− tan πx πx =1 secant power series, 330 Seidel’s, 349, 353 θ Seidel’s for log , 354 θ−1 4 403 −3 = 02 + 12 − 121+22 + 22−2 , +32 + 32 +42 399 −14 −24 −34 π2 = 02 +1 + 12 +22 + 22 +32 + 32 +42 , 399 µ(n) = − 212 − 312 · · · + n2 + · · · , 384 π Sondow’s, 247 √ (n!)2 22n π = limn→∞ (2n)! √n , 248 tangent continued fraction, 445 tangent power series, 329 Vi` ete’s, 154, 233, 237, 240 Wallis’, 234, 237, 247 µ(n) = ∞ n=1 nz , 375 ζ(z) π2 ζ(z) = 1− pz −1 , Proof I,II, 373 −1 , Proof III, 379 ζ(z) = − p1z τ (n) ∞ ζ(z) = n=1 nz , 380 ζ(2z) λ(n) = ∞ n=1 nz , 380 ζ(z) (k + 2) ζ(k + 1) = k−2 =1 ζ(k − ) ζ( + 1) + Hn , 265 ∞ n=1 nk ∞ ∞ ζ(k) = k−2 m=1 n=1 m (m+n)k− , =1 310 Fraction rules, 56 Function(s) absolute value, 17 additive, 171 bijective, 18 characteristic, 20 codomain of, 16 component, 161 composition of, 17 confluent hypergeometric limit function, 438 constant, 20 continuous, 166 cosecant, 200 cosine, 144, 198 cotangent, 200 definition, 16 Dirichlet, 17, 159 Dirichlet eta, 376 domain of, 16 equal, 20 even, 299 exponential, 94, 140, 187, 285 exponential, continued fraction, 442 graph of, 16 greatest integer, 69 Hurwitz zeta, 381 hyperbolic, 208 hyperbolic cosine, 200 hyperbolic sine, 200 hypergeometric, 438 identity, 20 image of, 16 injective, 18 inverse, 18 inverse or arc cosine (complex), 226 inverse or arc cosine (real), 218 inverse or arc sine (complex), 226 inverse or arc sine (real), 218 inverse or arc tangent (complex), 223 inverse or arc tangent (real), 218 inverse or arc tangent continued fraction, 398 jump, 186 Liouvilles, 380 logarithm, 189 Mă obius, 375, 379 monotone, 182 multiple-valued, 219 multiplicative, 171 nondecreasing, 182 odd, 299 one-to-one, 18 onto, 18 range of, 16 Riemann zeta, 192, 269, 285 Riemann zeta (in terms of Mă obius function), 375 Riemann zeta (in terms of primes), 373 secant, 200 sine, 144, 198 strictly decreasing, 182 strictly increasing, 182 surjective, 18 tangent, 200 target of, 16 the most important, 153, 187 value of, 16 Zeno, 182 Fundamental recurrence relations, 405 Fundamental theorem of algebra, 210 of algebra, proof I, 211 of algebra, proof II, 213 of algebra, proof III, 216 of arithmetic, 45 Galois, Evariste, 193, 452 Game coin, 34 of Nim, 40, 51 Towers of Hanoi, 33 Game of Nim, 40, 51 Gauss’ test, 293 Gauss, Carl, 10, 164 fundamental theorem of algebra, 210 INDEX on Borchardt’s algorithm, 235 quote, 81, 146, 199 Generalized power rules theorem, 190 Geometric mean, 34 Geometric series, 126 Geometric series theorem, 126 Gilfeather, Frank, 323 Glaisher, James quote, 340 Golden Ratio, 34 Golden ratio, 94, 113, 392, 414 continued fraction, 94, 123, 392 false rumors, 113, 392 infinite continued square root, 94, 113 the “most irrational number, 427 Good approximation, 423 Goodwin, Edwin, 228 Goto, Hiroyuki, 153 Graph, 16 Greatest common divisor, 46 Greatest integer function, 69 Greatest lower bound, 64 Gregorian calendar, 433 Gregory, James, 234, 338, 340 Gregory-Madhava’s arctangent series, 338 Hadamard, Jacques, 287 Halmos, Paul, Harmonic product, 351 series, 125, 130 Hermite, Charles, 271, 445 almost integer, 206 Heron of Alexandria, 79 Heron’s formula, 79 Hidden assumptions, 12 High school graph of the exponential function, 188 horizontal line test, 185 √ i = −1, 215 logarithms, 196 long division, 149 plane trigonometry facts, 207 trig identities, 199 zeros of (co)sine, 205 Hilbert, David, 389 Hobbes, Thomas, Hofbauer, Josef, 249, 251, 257 Holy bible, 153, 227, 423 House bill No 246, 153, 228 Hurwitz zeta function, 381 Hurwitz, Adolf, 381 Huygens, Christian, 433, 437 Hyperbolic cosine, 200 secant, power series of, 330 sine, 200 Hyperbolic cotangent 487 continued fraction, 439 Hyperbolic tangent continued fraction, 441 Hypergeometric function, 438 i, 81 I Kings, 227, 423 Identity Cassini, 411 Euler’s, 198 function, 20 Lagrange, 78 Pythagorean, 199 Identity theorem, 298 If then statements, 5, 11 If and only if statements, 13 II Chronicles, 227, 423 Image of a set, 18 Image of function, 16 Imaginary unit, 81 Independent events, 377 Index of a polynomial, 89 Induction, 23 Inductive definitions, 28 Inequalities rules, 38 strict, 23 Inequality Arithmetic-geometric mean, 34, 195 Bernoulli, 30 Cauchy-Bunyakovski˘ı-Schwarz, 74 Schwarz, 74 Inequality lemma, 174 Infimum, 64 Infinite countably, 83 interval, limits, 164 product, 240 series, 124 set, 83 uncountably, 83 Infinite product, 102 Injective, 18 inner product, 73 Inner product space, 74 Integer(s), 35 almost, 206 as a set, Intermediate value property, 176 Intermediate value theorem, 176 Intersection of sets, family of sets, Interval(s) end points, chains of, 181 closed, 488 infinite, left-half open, nested, 69 nontrivial, 170 of music, 435 open, right-half open, Inverse hyperbolic functions, 226 Inverse image of a set, 18 Inverse of a function, 18 Inverse or arc cosine (complex), 226 Inverse or arc cosine (real), 218 Inverse or arc sine (complex), 226 Inverse or arc sine (real), 218 Inverse or arc tangent (complex), 223 Inverse or arc tangent (real), 218 Irrational number, 55 Irrationality of er for r ∈ Q, 441 of e, proof I, 143 of e, proof II, 276 of e, proof III, 420 of log r for r ∈ Q, 446 of √ logarithmic numbers, 62 of 2, 57, 60, 63 of trigonometric numbers, 60, 63 Isolated point, 167 Iterated limits, 301 series, 145, 302 Jones, William, 153, 228 Julian calendar, 433 Jump, 183 discontinuity, 183 function, 186 Kanada, Yasumasa, 153, 234, 259, 340 Kasner’s number, 117 Kasner, Edward, 117 Khayyam, Omar, 434 King-Fang, 437 Knopp, Konrad, 344 Knott, Ron, 417 Kornerup, 437 Kortram, R.A., 372 k-th power free, 381 Kummer’s test, 290 Kummer, Ernest, 290 L’Hospital’s rule, 195 Lagrange identity, 78 Lagrange, Joseph-Louis, 450 Lambert, Johann, 153, 228, 441 Landau, Edmund, 291 Lange, Jerry, 399 Law of cosines, 78 Law of sines, 79 Leading coefficient, 59 INDEX Leap year, 433 Least upper bound, 63 Left-hand limit, 163 Lehmer, D.H., 342 Leibniz, Gottfried, 79, 234, 239, 340 function word, 15 on the series − + · · · , 93 quote, 435 Lemma Abel’s, 271 compactness, 173 connectedness, 174 inequality, 174 Length Euclidean, 74 of complex numbers, 82 Limit infimum, 281 supremum, 280 Limit comparison test, 137 lim inf, 281 Limit points and sequences lemma, 155 lim sup, 280 Limit(s) iterated, 301 left-hand, 163 of a function, 155 of a sequence, 95 open ball definition for functions, 156 open ball definition for sequences, 96 point, 154 right-hand, 163 Limits at infinity, 163 Lindemann, Ferdinand, 153, 228 linear polynomial, 88 Liouville number, 467 Liouville’s function, 380 Liouville’s theorem, 467 Liouville, Joseph, 380, 467 log 2, 154, 194 22 log = 11 11 , 396 + + 1+ ∞ log = n=2 2n ζ(n), 309 as alternating harmonic series, 154, 194 rearrangement, 311 Logarithm common, 62 complex, 221 power series, 336 function, 189 general bases, 196 natural, 189 Logarithmic test, 294 Logical quantifiers, 14 Lower bound, 64 greatest, 64 Lucas numbers, 418 INDEX Lucas, Franácois, 418 Mă obius function, 375 Mă obius inversion formula, 379 Mă obius, Ferdinand, 375 Machin’s formula, 342 Machin, John, 153, 228, 234, 342 Madhava of Sangamagramma, 234, 332, 338, 340 Massaging (an expression), 97, 117, 156, 157 Mathematical induction, 27 Max/min value theorem, 175 Maximum of a set, 70, 113 strict, 180 Mazur, Marcin, 63 Measurement of the circle, 226, 228 Meister, Gary, 323 Mengoli, Pietro, 249 Mercator, Gerhardus, 437 Mercator, Nicolaus, 332 Mersenne number, 48 Mersenne prime, 48 Mersenne, Marin, 48 Mertens’ multiplication theorem, 316 Mertens, Franz, 316 Method AGM (to compute π), 233 bisection, 181 of partial fractions, 125, 128, 210 Minimum of a set, 70 MIT cheer, 198 Mnemonic π, 153 Modular arithmetic, 47 Monotone criterion, 111 function, 182 sequence, 111 Monotone criterion theorem, 112 Monotone inverse theorem, 185 Monotone subsequence theorem, 113 “Most” extreme irrational, 468 important equation eiπ + = 0, 203 important function, 153, 187 irrational number, 427 real numbers are transcendental, 83 Multinomial theorem, 34 Multiple, 41 Multiple-valued function, 219 Multiplicative function, 171 Multiplicative identity existence for complex numbers, 80 existence for integers, 36 existence for natural numbers, 22 existence for reals, 54 489 vectors, 73 Multiplicative inverse existence for complex numbers, 80 existence for reals, 54 Multiplicity of a root, 88 Multiply by conjugate trick, 157 Music, 435 Nn , 83 n-th term test, 124 Natural numbers (N), 22 as a set, Negation of a statement, 14 Nested intervals, 69 Nested intervals theorem, 70 Newcomb, Simon, 226 Newton, Sir Isaac, 332 Niven, Ivan, 322 Nondecreasing function, 182 sequence, 111 Nonincreasing function, 182 sequence, 111 Nonnegative series test, 125 Nontrivial interval, 170 Norm, 76 ball, 76 box, 79 Euclidean, 74 sup (or supremum), 79 Normed space, 76 Null sequence, 96 Number theorem series, 310 Number(s) algebraic, 88 approximable, 464 Bernoulli, 262, 270, 327 composite, 44 coprime, 47, 153, 234, 378 Euler, 270, 330 extended real, 109, 280 Liouville, 467 perfect, 48 prime, 44 relatively prime, 47, 234, 378 square-free, 153, 322 transcendental, 88 Odd number, 44 function, 299 1/n-principle, 67 One-to-one, 18 Onto, 18 Open ball, 77 interval, set, 173 490 Order laws integers, 38 natural numbers, 22 reals, 55 Ordered field, 54 Orthogonal, 77 Oscillation theorem, 203 p-series, 133, 134 p-test, 134 Pandigital, 432 Pappus of Alexandria, 456 Paradox Russell, Bertrand, 11 Vredenduin’s, 90 Parallelogram law, 77 Partial products, 351 sum, 124 Partial quotients, 416 Pascal’s method, 34 Pascal’s rule, 33 Pascal, Blaise, 349 Peirce, Benjamin, 203 Pell equation, 458 Pell, John, 459 Perfect number, 48 Period of a b-adic expansion, 149 of a continued fraction, 447 Periodic b-adic expansions, 149 continued fraction, 447 purely periodic continued fraction, 452 Persian calendar, 437 Pfaff, Johann, 235 Φ, see also Golden ratio π, 198, 226 and the unit circle, 204 continued fraction, 400, 419 definition of, 202 formulas, 233 origin of letter, 228 Vi` ete’s formula, 154, 233, 237 Pianos, 435 Pigeonhole principle, 84 Pochhammer symbol, 438 Pochhammer, Leo August, 438 Poincar´ e, Henri, 26, 189 Point isolated, 167 limit, 154 Pointwise discontinuous, 170 Polar decomposition, 83 Polar representation, polar coordinates, 206 Polynomial complex, 87 degree of, 59 INDEX index of, 89 leading coefficient of, 59 linear, 88 Pope Gregory XIII, 433 Positivity property integers, 36 reals, 54 Power complex, 190, 222 rules (generalized), 190 rules (integer powers), 57 rules (rational powers), 67 Power series, 287, 295 composition of, 325 Power series composition theorem, 325 Power series division theorem, 327 Preimage of a set, 18 Preservation of inequalities theorem limits of functions, 162 limits of sequences, 106 Prime(s) definition, 44 infinite series of, 270, 320 infinitude, 44 sparseness, 46 Primitive Pythagorean triple, 47, 432 Principal argument, 219 inverse hyperbolic cosine, 226 inverse hyperbolic sine, 226 inverse or arc cosine, 226 inverse or arc sine, 226 inverse or arc tangent, 223 logarithm, 221 n-th root, 215 value of az , 222 Principle ε, 102 of mathematical induction, 27 1/n, 67 pigeonhole, 84 Principle of mathematical induction, 27 Pringsheim’s theorem for double sequences, 302 for double series, 302 for sequences, 137 Pringsheim, Alfred, 302 Probability, 377 number is square-free, 234, 378 numbers being coprime, 234, 378 Product Cartesian, 15 Cauchy, 315 of sequences, 108 Proof by cases, 39 contradiction, 25 contrapositive, 24, 39 INDEX direct, 24 Properly divergent functions, 164 sequences, 109 Property(ies) intermediate value, 176 of Arctan, 224 of lim inf/sup, 281 of lim inf/sup theorem, 281 of power series, 295 of sine and cosine, 199 of the complex exponential, 141 of the logarithm, 189 of the real exponential, 188 of zero and one theorem, 39 Purely periodic continued fraction, 452 Puzzle antipodal points on earth, 180 barber, bent wire puzzle, 179 coconut, 409 irrational-irrational, 195 mountain, 178 numbers one more than their cubes, 179 rational-irrational, 192 square root, 57 Which is larger, π e or eπ ?, 206 Pythagorean identity, 199 Pythagorean theorem, 77 Pythagorean triple, 47, 432, 463 Quadratic irrational, 448 Quote Abel, Niels, 123 Bernoulli, Jacob, 125, 269 Brahmagupta, 459 Cantor, Georg, 83 d’Alembert, Jean Le Rond, 95, 284 Dirac, Paul, 237 Eddington, Sir Arthur, 21 Euler, Leonhard, 41, 93 Galois, Evariste, 193 Gauss, Carl, 10, 81, 146, 164, 199 Glaisher, James, 340 Halmos, Paul, Hermite, Charles, 271 Hilbert, David, 389 Hobbes, Thomas, Leibniz, Gottfried, 79, 435 MIT cheer, 198 Newcomb, Simon, 226 Peirce, Benjamin, 203 Poincar´ e, Henri, 26, 189 Ramanujan, Srinivasa, 432 Richardson, Lewis, 11 Russell, Bertrand, 11 Schră odinger, Erwin, 55 Smith, David, 153 491 Wigner, Eugene, 79 Zagier, Don Bernard, 44 Quotient, 41, 42 of sequences, 108 Quotients complete, 416 partial, 416 Raabe’s test, 291 Raabe, Joseph, 291 Radius of convergence, 287 of (co)tangent, 388 Ramanujan, Srinivasa approximation to π, 432 quote, 432 Range of function, 16 Ratio comparison test, 138 Ratio test for sequences, 111 for series, 284 Rational numbers as a set, Rational numbers (Q), 54 Rational zeros theorem, 59 Real numbers (R), 53 Rearrangement, 311 Recurrence relations fundamental, 405 Wallis-Euler, 404 Reductio ad absurdum, 25 relatively prime numbers, 47, 234, 378 Remainder, 42 Remmert, Reinhold, 210 Rhind (or Ahmes) papyrus, 227 Rhind, Henry, 227 Richardson, Lewis, 11 Riemann zeta function, 192, 269, 285, 373, 375 infinite product, 350 Riemann’s rearrangement theorem, 312 Right-hand limit, 163 Ring, 36 Root of a complex number, 210 of a real number, 57 or zero, 178 principal n-th, 215 rules, 66 Root test for sequences, 110 for series, 284 Roots of polynomials, 87 Rule L’Hospital’s, 195 Rules absolute value, 39 divisibility, 42 fraction, 56 492 inequality, 38 of sign, 37 power (general), 190 power (integers), 57 power (rational), 67 root, 66 Russell’s paradox, 11 Russell, Bertrand barber puzzle, paradox, 11 quote, 11 s-adic expansions, 151 Salamin, Eugene, 233 Schră odinger, Erwin, 55 Schwartz inequality, 77 Schwarz inequality, 74 Schwarz, Hermann, 74 Searc´ oid, M´ıche´ al, 213 Secant function, 200 power series, 330 Seidel, Ludwig, 349, 353 Semiperimeter, 79 Septimal system, 50 Sequence, 17 Cauchy, 117 contractive, 120 definition, 94 difference, 122 double, 145, 300 Fibonacci, 34, 131, 289, 341, 411 Lucas, 418 monotone, 111 nondecreasing, 111 nonincreasing, 111 null, 96 of bounded variation, 272 of sets, strictly decreasing, 114 strictly increasing, 114 subsequence of, 106 tail, 103 Sequence criterion for continuity, 167 for limits of functions, 159 Series absolute convergence, 135 alternating harmonic, 136 binomial, 333 geometric, 126 Gregory-Leibniz-Madhava, 239, 340 harmonic, 125, 130 involving number and sum of divisors, 310 iterated, 145, 302 Leibniz’s, 239, 340 power, 287, 295 telescoping, 128 Set(s) INDEX compact, 173 complement of, connected, 173 countable, 83 countably infinite, 83 cover of, 172 definition, dense, 170 difference of, disconnected, 173 disjoint, empty, family of, finite, 83 image of, 18 infimum of, 64 infinite, 83 intersection of, inverse image of, 18 least upper bound of, 63, 64 limit point of, 154 lower bound of, 64 maximum of, 113 open, 173 preimage of, 18 supremum of, 63 uncountable, 83 union of, upper bound of, 63 Venn diagram of, zero, 171 Shanks, William, 234 Sharp, Abraham, 343 Sine function, 144, 198 Smith, David, 153 Somayaji, Nilakantha, 340 Sondow, Jonathan, 247 Square root, 57 Square-free numbers, 153, 234, 322, 378 Squaring the circle, 153, 228 Squeeze theorem functions, 162 sequences, 105 Standard deviation, 463 Stirling’s formula, weak form, 115 Strict maxima, 180 Strictly decreasing sequence, 114 Strictly increasing function, 182 sequence, 114 Subcover, finite, 172 Subsequence, 106 Subset, Sum by curves theorem, 304 1/p, 320 Summation arithmetic progression, 29 INDEX by curves, 304 by squares, 303 by triangles, 304 definition, 28 geometric progression, 30 of powers of integers, 272, 331 Pascal’s method, 34 Summation by parts, 271 Sup (or supremum) norm, 79 Supremum, 63 Surjective, 18 Tøndering, Claus, 433 Tail of a sequence, 103 Tails theorem for sequences, 103 Tails theorem for series, 126 tangent continued fraction, 445 Tangent function, 200 power series of, 329 Tannery’s theorem for products, 359 Tannery’s theorem for series, 138 Tannery, Jules, 138 Target of function, 16 Telescoping series, 128 Telescoping series theorem, 128 generalization, 131 Temperment, 436 Terminating decimal, b-adic expansion, 148 Tertiary system, 51 Test(s) Abel’s, 277 alternating series, 275 Cauchy condensation, 133 comparison, 132 De Morgan and Bertrand’s, 293 Dirichlet, 273 Gauss’, 293 Kummer’s, 290 limit comparison, 137 logarithmic, 294 n-th term test, 124 nonnegative series test, 125 p-test, 134 Raabe’s, 291 ratio comparison, 138 ratio for sequences, 111 ratio for series, 284 root for sequences, 110 root for series, 284 The Sand Reckoner, 456 Theorem Abel’s limit, 296 Abel’s multiplication, 317 Basic properties of sine and cosine, 199 best approximation, 425, 429 binomial, 31 493 Bolzano-Weierstrass, 114 boundedness, 175 Cauchy’s arithmetic mean, 344 Cauchy’s double series, 145, 305 Cauchy’s multiplication, 318 Cauchy-Hadamard, 287 Continued fraction convergence, 416 continuity of power series, 295 Dirichlet for rearrangements, 314 Dirichlet’s approximation, 430 existence of complex n-th roots, 215 FTA, proof I, 211 FTA, proof II, 213 FTA, proof III, 216 generalized power rules, 190 identity, 298 intermediate value, 176 Liouville’s, 467 max/min value, 175 Mertens’ multiplication, 316 monotone criterion, 112 monotone inverse, 185 monotone subsequence, 113 oscillation, 203 π and the unit circle, 204 power series composition, 325 power series division, 327 Pringsheim’s, for double sequences, 302 Pringsheim’s, for double series, 302 Pringsheim’s, for sequences, 137 properties of the complex exponential, 141 properties of the logarithm, 189 properties of the real exponential, 188 Riemann’s rearrangement, 312 sum by curves, 304 Tannery’s for products, 359 Tannery’s for series, 138 Thomae’s function, 169, 186 Thomae, Johannes, 169 Towers of Hanoi, 33 Transcendental number, 88 Transformation rules, 394 Transitive law cardinality, 84 for inequalities, 23 for sets, Triangle inequality for Rm , 75 for integers, 40 for series, 135 Trick(s) divisibility, 53 ε/3, 277 ε/2, 102, 119 multiply by conjugate, 157 probability that a number is divisible by k, 378 to find N , 97 494 Trisect an angle, 228 Tropical year, 433 2-series, see also Euler’s sum for π /6 Unary continued fraction, 422 Uncountability of transcendental numbers, 89 of irrational numbers, 87 of real numbers, 87 Uncountable, 83 Union of sets, family of sets, Uniqueness additive identities and inverses for Z, 37 multiplicative inverse for R, 56 of limits for functions, 160 of limits for sequences, 102 Upper bound, 63 least, 63 Value of function, 16 Vanden Eynden, Charles, 325 Vardi, Ilan, 339, 341 Vector space, 73 Vectors, 72 Venn diagram, Venn, John, Vi` ete, Fran¸cois, 154, 233, 349 Volterra’s theorem, 170 Volterra, Vito, 170 Vredenduin’s paradox, 90 Waldo, C.A., 228 Wallis’ formulas, 247 Wallis, John, 234, 237, 247 infinity symbol ∞, Wallis-Euler recurrence relations, 404 Wantzel, Pierre, 228 Weierstrass, Karl, 114 Weinstein, Eric, 432 Weisstein, Eric, 247 Well-ordering (principle) of N, 25 Wigner, Eugene, 79 Wiles, Andrew, 462 Williams’ formula, 259 Williams’ other formula, 265 Williams, G.T., 258 Yasser, 437 Zagier, Don Bernard, 44 Zeno of Elea, 182 Zeno’s function, 182 Zero of a function, 178 set of a function, 171 Zeta function, 134, see also Riemann zeta function INDEX ... between set theory and math statements/proofs • Define functions and the operations of functions on sets SETS, FUNCTIONS, AND PROOFS 1.1 The algebra of sets and the language of mathematics In... approximations and the mystery of π solved! 8.6 Continued fractions and calendars, and math and music 8.7 The elementary functions and the irrationality of ep/q 8.8 Quadratic irrationals and periodic... convergence for sequences and series and recognize monotone and Cauchy sequences • apply the rigorous ε-δ definition of limits for functions and continuity and the fundamental theorems of continuous functions