Sequence and Series
Author: Zhiming Ou
Mattermatics Learning Center
Publisher: 3265 Public Way
ISBN: 978-0-9866371-9-3
Summary
Starting from the simple pattern in a sequence of numbers, we travel through arithmetic sequence, geometric sequence and general recursive sequence and series, and delve into infinite sequences and series. Many natural phenomena can only be described by an infinite process.
It is an important ability to discover the pattern involved in a list of objects. The general methods are (1) comparing consecutive terms, (2) by mathematical induction, (3) solving mathematical recursion, (4) finding a closed form for a series. This book shows all the skills to obtain this ability.
Content
Chapter 1 Sequence of Numbers 4
- 1.1 General term of a Sequence 4
- 1.2 Patterns in Equations 12
Chapter 2 Arithmetic Sequences and Series 14
- 2.1 Arithmetic Sequence 14
- 2.2 Arithmetic Series 17
Chapter 3 Geometric Sequence and Series 21
- 3.1 Geometric Sequences 21
- 3.2 Geometric Series 23
Chapter 4 Financial Mathematics 26
- 4.1. Compounded Interest 26
- 4.2. Present Values 27
- 4.3. Amount of Annuity 28
- 4.4. Present Value of an Annuity 29
- 4.5. Installments 30
Chapter 5 Recursive Sequences 33
- 5.1 Concept of Recursive Sequence 33
- 5.2 Solving Recursive Sequences 36
- 5.3 Series and Summation 40
Sigma Notation 40
The Telescoping Skill 41
Relationship between Sn and tn 42
Chapter 6 Mathematical Induction 43
- 6.1 The Principle of Mathematical Induction 43
Proof by Induction 44
The Principle of Mathematical Induction 46
- 6.2 The Method of Induction 49
- 6.3 Flawed Proofs by Induction 52
Chapter 7 Transformations of Sequences 54
- 7.1 Transform a Sequence to some known Sequence 54
- 7.2 Recursive equations with complicated functions 55
- 7.3 Properties of Sequences 56
Chapter 8 Advanced Methods for Solving Linear Recurrence 59
- 8.1 Formula for General Linear Recurrence 60
- 8.2 Infinite Geometric Series 61
- 8.3 Generating Functions 62
Chapter 9 Infinite Series and Products 65
- 9.1 Concept 65
- 9.2 Linear Operations with Series 68
- 9.3 Test of Convergence 70
Chapter 10 Power Series 81
- 10.1 Convergence 81
- 10.2 Representations of Functions as Power Series 84
- 10.3 Operations on Power Series 88
Chapter 11 Trigonometric Series 92
- 11.1 Convergence of Trigonometric Series 92
- 11.2 Fourier Series 93
Convergence 95
Parseval’s Identity 96
Poisson’s Summation Formula 96
Chapter 12 General Function Term Series 98
- 12.1 Definition 98
uniformly convergent series 99
- 12.2 Infinite Products 100
- 12.3 Dirichlet Series 101