Sequences and Power Series
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About the book
Description
A book with 'Guidelines for Solutions of Problems'. The book is in English.
Preface
Here follow some guidelines for solution of problems concerning sequences and power series. It should be emphasized that my purpose has never been to write an alternative textbook on these matters. If I would have done so, I would have arranged the subject differently. Nevertheless, it is my hope that the present text can be a useful supplement to the ordinary textbooks, in which one can find all the necessary proofs which are skipped here.
The text presupposes some knowledge of Calculus 1a, Functions in One Variable, and it will itself be the basis for the following Calculus 4b: Fourier Series, Differential Equations and Eigenvalue Problems. The previous text, Calculus 2b, Functions in Several Variables will only be necessary occasionally.
Chapter 1 is a repetition of useful formulæ – some of them already known from high school – which will be used over and over again. The reader should read this chapter carefully together with Appendix A, which is a short collection of formulæ known previously. These will be assumed in the text without further reference, so it would be a good idea to learn these formulæ by heart, since they can be considered as the tools of Calculus which should be mastered before one can proceed.
The text itself falls into two main parts, 1) Sequences of numbers and functions, and 2) Series of numbers and power series. The more general series of functions occur only rarely in this text. I felt that the main case of Fourier series should be put into a later text, because the natural concept of convergence is not the same as the convergence dealt with here. I have seen too many students being confused by the different types of convergence to let these two main cases clash in the same volume.
Comments, remarks and examples will always be ended by the symbol <>, so the reader can see when the main text starts again.
In general, every text in the Calculus series is given a number – here 3 – and a letter – here b – where
a means “compendium”,
b means “guidelines for solutions of standard problems”,
c means “examples”.
Since this is the first edition of this text, there may still be some errors, which the reader hopefully will forgive me.
Leif Mejlbro
Content
 Repetition of important formulæ
 Decomposition
 Trigonometric formulæ
 Notations and conventions
 Standard power series
 Power like standard series3
 Recognition of power like series
 Exponential like standard series
 Recognition of exponential like series
 Integration of trigonometric polynomials
 Use of pocket calculators
 Real sequences, folklore
 Rules of magnitude
 Square roots etc.
 Taylor’s formula
 Standard sequences
 Practical methods for sequences
 Sequences
 Iterative sequences
 Sequences of functions
 General series; methods in problems
 Denition
 Rules of calculus
 Change of index
 A general advice
 Elementary standard series
 Types of Convergence
 An elaboration on the ow diagram
 Convergence tests
 Series of functions
 Power series; methods in solution of problems
 Standard power series
 The structure of standard series
 Convergence of power series
 Review of some important theorems
 Sum by termwise differentiation or integration
 The method of power series
 Recursion formulæ and difference equations
 Second order differential equations (straight tips)
 Differential equation of second order
 Formulæ
 Squares etc.
 Powers etc.
 Differentiation
 Special derivatives
 Integration
 Special antiderivatives
 Trigonometric formulæ
 Hyperbolic formulæ
 Complex transformation formulæ
 Taylor expansions
 Magnitudes of functions