附註:Includes bibliographical references (pages 346-349) and index.
Cover -- Half-title -- Series-title -- Title -- Copyright -- Contents -- Preface to first edition -- Preface to second edition -- Glossary -- PART I Numbers -- 1 Mathematical induction -- Historical note -- Answers and comments -- 2 Inequalities -- Positive numbers and their properties -- Summary -- properties of order -- Arithmetic mean and geometric mean -- Completing the square -- The sequence (1+ 1/n) -- nth roots -- Summary -- results on inequalities -- Absolute value -- Summary -- results on absolute value -- Historical note -- Answers and comments -- 3 Sequences A first bite at infinity.
Monotonic sequences -- Bounded sequences -- Subsequences -- Sequences tending to infinity -- Archimedean order and the integer function -- Summary -- the language of sequences -- Null sequences -- Summary -- null sequences -- Convergent sequences and their limits -- Boundedness of convergent sequences -- Quotients of convergent sequences -- d'Alembert's ratio test -- Convergent sequences in closed intervals -- Intuition and convergence -- Summary -- convergent sequences -- Historical note -- Answers and comments -- 4 Completeness What the rational numbers lack.
The Fundamental Theorem of Arithmetic -- Dense sets of rational numbers on the number line -- Infinite decimals -- Irrational numbers -- Infinity: countability -- Summary -- Denseness -- Decimals and irrationals -- Countability -- The completeness principle: infinite decimals are convergent -- Every infinite decimal sequence is convergent. -- Bounded monotonic sequences -- nth roots of positive real numbers, n a positive integer -- Nested closed intervals -- Convergent subsequences of bounded sequences -- Cluster points (the Bolzano-Weierstrass theorem) -- Cauchy sequences.
Least upper bounds (sup) and greatest lowest bounds (inf) -- Upper bounds and greatest terms -- Least upper bound (sup) -- Lower bounds and least members -- Greatest lower bound (inf) -- sup, inf and completeness -- lim sup and lim inf -- Summary -- completeness -- Historical note -- Answers and comments -- 5 Series Infinite sums -- Sequences of partial sums -- The null sequence test -- Simple consequences of convergence -- Summary -- convergence of series -- Series of positive terms -- First comparison test -- The harmonic series -- The convergence of Sigma 1/n -- Cauchy's nth root test.
D'Alembert's ratio test -- Second comparison test -- Integral test -- Summary -- series of positive terms -- Series with positive and negative terms -- Alternating series test -- Absolute convergence -- Conditional convergence -- Rearrangements -- Summary -- series of positive and negative terms -- Power series -- Applications of d'Alembert's ratio test and Cauchy's nth root test for absolute convergence -- Radius of convergence -- Cauchy-Hadamard formula -- The Cauchy product -- Summary -- power series and the Cauchy product -- Historical note -- Answers and comments -- PART II Functions.
摘要:The transition from studying calculus in schools to studying mathematical analysis at university is notoriously difficult. In this second edition of Numbers and Functions, Dr Burn invites the student reader to tackle each of the key concepts in turn, progressing from experience through a structured sequence of several hundred problems to concepts, definitions and proofs of classical real analysis. The sequence of problems, which all have solutions supplied, draws students into constructing definitions and theorems for themselves. This natural development is informed and complemented by historical insight. The novel approach to rigorous analysis offered here is designed to enable students to grow in confidence and skill and thus overcome the traditional difficulties.
