Schinazi R.B. From Calculus to Analysis
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Rinaldo B. Schinazi
From Calculus
to Analysis
Rinaldo B. Schinazi
Department of Mathematics
University of Colorado
Colorado Springs, CO 80933-7150
USA
rschinaz@uccs.edu
ISBN 978-0-8176-8288-0 e-ISBN 978-0-8176-8289-7
DOI 10.1007/978-0-8176-8289-7
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2011938703
Mathematics Subject Classification (2010): 26-01
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To Tina, with love
Preface
I have taught elementary analysis many times. It is a difficult course for students
and instructor alike. Students lack basic techniques such as manipulations of simple
inequalities with elementary functions. For instance, most students will have trouble
finding upper and lower numerical bounds for
1
1+x
2
when −4 <x<−2; they also
lack mathematical culture at this stage. Many students have no idea why we need
limits, why series are important, what the number π is, and so on. These gaps are
actually not that surprising given how little theory students have been exposed to up
to that point.
To decrease the failure rate in analysis and make it a less traumatic experience
for both the students and the instructor, we have offered a pre-analysis course for
the last several years at the University of Colorado at Colorado Springs. Students
are strongly advised to take one semester of pre-analysis, and they then take the
required one semester of analysis. The experiment has been remarkably successful.
The failure rate in analysis has dropped significantly, and I get many more positive
comments about the whole learning experience. The only problem is that there are
rather few textbooks available for a pre-analysis course. The main goal of this work
is to provide such a textbook.
I use Chaps. 1 through 4 for my pre-analysis course. My goal is to get students
comfortable at estimating simple algebraic expressions and at the same time in-
crease their mathematical culture. In order to achieve these two goals simultane-
ously, the order in which topics appear is not the traditional one, and many concepts
are introduced several times. For instance, I compute derivatives in Chap. 3 long
before differentiation is defined in Chap. 5.
I have also used my notes that evolved into this book for a classical elementary
analysis course (1.3, 2.1, 2.2, Chaps. 5 and 6, and selected topics from Chaps. 7, 8,
and 9). Chapter 7 is a short introduction to uniform convergence. It also contains
the proofs of the classical results on differentiation and integration of power series
that are used in Chaps. 3 and 4. Chapters 8 and 9 are introductions to decimal rep-
resentations of the reals and countability, respectively. Chapter 8 is the closest we
get to constructing the reals, and Chap. 9 contains important ideas and results that
vii
viii Preface
students need to be exposed to at this point in their mathematical education and ca-
reer. The chapters of the book are largely independent from each other, and the only
prerequisite is a standard calculus course.
Rinaldo B. SchinaziColorado Springs, CO, USA
Contents
1 Number Systems ............................. 1
1.1 The Algebra of the Reals . . . ................... 1
1.2 NaturalNumbersandIntegers.................... 6
1.3 Rational Numbers and Real Numbers ................ 13
1.4 Power Functions ........................... 23
2 Sequences and Series ........................... 33
2.1 Sequences . ............................. 33
2.2 Monotone Sequences, Bolzano–Weierstrass Theorem, and
OperationsonLimits ........................ 40
2.3 Series................................. 45
2.4 Absolute Convergence ........................ 55
3 Power Series and Special Functions ................... 65
3.1 PowerSeries............................. 65
3.2 Trigonometric Functions ....................... 67
3.3 Inverse Trigonometric Functions .................. 79
3.4 Exponential and Logarithmic Functions ............... 82
4 Fifty Ways to Estimate the Number π ................. 95
4.1 Power Series Expansions . . . ................... 95
4.2 Wallis’ Integrals, Euler’s Formula, and Stirling’s Formula .....108
4.3 Convergence of Infinite Products ..................120
4.4 The Number π Is Irrational . . ...................131
5 Continuity, Limits, and Differentiation .................137
5.1 Continuity ..............................137
5.2 Limits of Functions and Derivatives .................147
5.3 Algebra of Derivatives and Mean Value Theorems .........157
5.4 Intervals, Continuity, and Inverse Functions .............167
6 Riemann Integration ...........................177
6.1 ConstructionoftheIntegral .....................177
ix