A First Course in Mathematical Analysis

by David Alexander Brannan

• ISBN13: 9780521864398
• ISBN10: 0521864399
• Format: Hardcover
• Copyright: 2006-09-04
• Publisher: Cambridge University Press

Summary

Mathematical Analysis (often called Advanced Calculus) is generally found by students to be one of their hardest courses in Mathematics. This text uses the so-called sequential approach to continuity, differentiability and integration to make it easier to understand the subject. Topics that are generally glossed over in the standard Calculus courses are given careful study here. For example, what exactly is a 'continuous' function? And how exactly can one give a careful definition of 'integral'? The latter question is often one of the mysterious points in a Calculus course - and it is quite difficult to give a rigorous treatment of integration! The text has a large number of diagrams and helpful margin notes; and uses many graded examples and exercises, often with complete solutions, to guide students through the tricky points. It is suitable for self-study or use in parallel with a standard University course on the subject.

Table of Contents

Preface	ix
Numbers	1	(36)
Real numbers	2	(7)
Inequalities	9	(5)
Proving inequalities	14	(8)
Least upper bounds and greatest lower bounds	22	(8)
Manipulating real numbers	30	(5)
Exercises	35	(2)
Sequences	37	(46)
Introducing sequences	38	(5)
Null sequences	43	(9)
Convergent sequences	52	(9)
Divergent sequences	61	(7)
The Monotone Convergence Theorem	68	(11)
Exercises	79	(4)
Series	83	(47)
Introducing series	84	(8)
Series with non-negative terms	92	(11)
Series with positive and negative terms	103	(19)
The exponential function x → ex	122	(5)
Exercises	127	(3)
Continuity	130	(37)
Continuous functions	131	(12)
Properties of continuous functions	143	(8)
Inverse functions	151	(10)
Defining exponential functions	161	(3)
Exercises	164	(3)
Limits and continuity	167	(38)
Limits of functions	168	(8)
Asymptotic behaviour of functions	176	(5)
Limits of functions -- using ε and δ	181	(12)
Continuity -- using ε and δ	193	(7)
Uniform continuity	200	(3)
Exercises	203	(2)
Differentiation	205	(50)
Differentiable functions	206	(10)
Rules for differentiation	216	(12)
Rolle's Theorem	228	(4)
The Mean Value Theorem	232	(6)
L'Hospital's Rule	238	(6)
The Blancmange function	244	(8)
Exercises	252	(3)
Integration	255	(58)
The Riemann integral	256	(16)
Properties of integrals	272	(10)
Fundamental Theorem of Calculus	282	(6)
Inequalities for integrals and their applications	288	(15)
Stirling's Formula for n!	303	(6)
Exercises	309	(4)
Power series	313	(41)
Taylor polynomials	314	(6)
Taylor's Theorem	320	(9)
Convergence of power series	329	(9)
Manipulating power series	338	(8)
Numerical estimates for π	346	(4)
Exercises	350	(4)
Appendix 1 Sets, functions and proofs	354	(5)
Appendix 2 Standard derivatives and primitives	359	(2)
Appendix 3 The first 1000 decimal places of √2, e and π	361	(2)
Appendix 4 Solutions to the problems	363	(94)
Chapter 1	363	(8)
Chapter 2	371	(11)
Chapter 3	382	(11)
Chapter 4	393	(9)
Chapter 5	402	(11)
Chapter 6	413	(13)
Chapter 7	426	(17)
Chapter 8	443	(14)
Index	457