Problems and Methods in Analysis

ForewordVolume 11. Infinite Sequences § 1. General Remarks § 2. Theorems and Properties Exercise 12. Number Series § 1. General Remarks § 2. Theorems and Properties 2.1 Necessary Condition for Convergence 2.2 Comparison Test 2.3 The Geometric Series 2.4 The Harmonic Series 2.5 The Harmonic Series of Order a 2.6 Two Groups of Series § 3. Series of Non-Negative Terms 3.1 Criterion for Convergence 3.2 Criterion for Divergence 3.3 D'Alembert's Ratio Test 3.4 Cauchy's Test § 4. Alternating Series 4.1 Leibniz's Test 4.2 Absolute Convergence § 5. Other Series Exercise 23. Functions and their Derivatives § 1. Definition of a Function § 2. Limit of a Function § 3. Continuity 3.1 Properties of Continuous Functions 3.2 Examples of Continuous Functions Exercise 3 § 4. First Order Derivatives of Functions of One Variable 4.1 Definition 4.2 Definition 4.3 Geometrical Interpretation 4.4 Differentiability of Continuous Functions 4.5 Properties of First Derivatives 4.6 Function of a Function 4.7 Inverse Functions 4.8 A List of First Derivatives Exercise 4 § 5. Derivatives of Higher Order 5.1 Definition 5.2 Derivatives of Order n Exercise 5 § 6. Derivatives of a Function Given by Parametric Equations 6.1 First Derivatives 6.2 Second Derivatives Exercise 64. Partial Differentiation § 1. Continuity of a Function of Two Variables § 2. First Order Derivatives § 3. Second and Higher Order Derivatives 3.1 Introduction 3.2 Theorem 3.3 Definition 3.4 Function of a Function 3.5 Functions of Several Variables Exercise 7 § 4. Derivatives of Implicit Functions 4.1 Definition 4.2 The Existence of Implicit Functions Exercise 85. Algebra § 1. Complex Numbers 1.1 Definitions 1.2 Trigonometric Interpretation 1.3 De Moivre's Theorem Exercise 9 § 2. The Solution of Algebraic Equations 2.1 General Properties of Algebraic Equations Exercise 10 2.2 Cubic Equations Exercise 11 2.3 Quartic Equations Exercise 126. Curve Tracing §1. Maxima and Minima 1.1 Increasing and Decreasing Functions 1.2 Definition of Maxima and Minima 1.3 Stationary Points 1.4 Turning Points 1.5 Determination of the Nature of Turning Points § 2. Concavity 2.1 Definition 2.2 Points of Inflexion § 3. Asymptotes 3.1 Definition 3.2 Asymptotes Parallel to the Coordinate Axes 3.3 Oblique Asymptotes §4. Curve Tracing 4.1 Procedure 4.2 General Notes § 5. Implicit Functions 5.1 Definition 5.2 Conditions for a Multiple Point 5.3 Tangents at Multiple Points 5.4 Two Rules for Finding Asymptotes 5.5 Technique § 6. Functions of the type yq = xp 6.1 Symmetry 6.2 p/q > 1 6.3 p/q Exercise 137. Power Series § 1. General Remarks 1.1 Definition 1.2 Radius of Convergence 1.3 Theorems Exercise 14 § 2. Taylor's Theorem 2.1 Definition 2.2 The Taylor Series of a Function 2.3 The Maclaurin Series of a Function 2.4 The Integration of Any Series Exercise 158. Limiting Values of Indeterminate Forms § 1. The Indeterminate Symbols 0/0, 8/8 1.1 The rule of 1'Hôpital 1.2 Generalization of Rule § 2. The Indeterminate Symbols 0.8, 8 - 8, 18, 8°, 0° 2.1 Theorem (0.8) 2.2 Theorem (8 - 8) 2.3 Theorem (18, 8°, 0°) Exercise 169. Approximation to Roots of Equations § 1.