Math 327A Exercises 4. Due Wednesday, February 3, 2010.
Problems from the book, chapter 16.
Deﬁnition: Consider some set S of real numbers. A real number a is an accumulation point
of the set S if for each positive value ǫ, there is a number xǫ in S (othe
Math 327A Exercises 4. Due Wednesday, Feb 3, 2010.
Problems from the book, chapter 16.
Deﬁnition: Consider some set S of real numbers. A real number a is an accumulation point
of the set S if for each positive value ǫ, there is a number xǫ in S (other tha
Math 327A. Exercise 3, due Wednesday, January 27, 2010.
An increasing sequence {an }∞ converges if and only if the set of terms of the sequence
n=1
has an upper bound. If the set of terms of the sequence has an upper bound, the sequence
converges to the l
Math 327A. Exercise 3, due Wednesday, January 27, 2010.
An increasing sequence {an }∞ converges if and only if the set of terms of the sequence
n=1
has an upper bound. If the set of terms of the sequence has an upper bound, the sequence
converges to the l
Math 327A Exercise 2. Due Wednesday, January 20, 2010.
1
1. The decreasing sequence { n }∞ converges to 0. What can you say about the sequence
n=1
{1/3, 1/2, 1, 1/6, 1/5, 1/4, 1/9, 1/8, 1/7, .}?
Solution: The sequence is not decreasing, but its limit is s
Math 327A Exercise 2. Due Wednesday, January 20, 2010.
1
1. The decreasing sequence { n }∞ , i.e., the decreasing sequence 1,1/2.1/3,1/4,1/5,1/6,.,1/n,.
n=1
converges to 0. The sequence {1/3, 1/2, 1, 1/6, 1/5, 1/4, 1/9, 1/8, 1/7, .} is not decreasing,
but
Math 327A Exercise 1. Due Monday, January 11, 2010.
From the beginning, it should be understood that limt→∞
t ranges over the positive integers.
1
t
= 0. In that equality, the variable
Let {an }∞ be an inﬁnite sequence of real numbers, and let m be a real
Math 327A Exercise 1. Due Monday, January 11, 2010.
From the beginning, it should be understood that limt→∞
t ranges over the positive integers.
1
t
= 0. In that equality, the variable
Let {an }∞ be an inﬁnite sequence of real numbers, and let m be a real
Math 327A. Sample midterm problems. February 5, 2010.
1
1. Check that the series Σ∞ n ln2 n converges and give the best upper bound on its value
n=2
that you can.
2. The ratio test is disappointing for Σ∞ an when limn→∞ an+1 = 1, because it gives
n=0
an
MATH 327 Winter 2010
Instructor: John Sullivan
Oﬃce: Padelford C-341. Phone 543-7986.
E-mail: [email protected]
Oﬃce hours: M 1:45-2:45, Tu 1:30-2:30 or by appt.
Web page: www.math.washington.edu/ sullivan/personal.html
Text: Advanced Calculus,
Math 444
Geometry for Teachers
Homework Expectations
Winter 2010
Due Date: Each written assignment has a due date; the assignment should be turned in at the beginning
of class on that day. Homework turned in after the ﬁrst ten minutes of class will get a
Math 444
Geometry for Teachers
Winter 2010
Some Challenge Problems
1. In the following diagram, AB = BC = CD and AD = BD. Find the measure of angle D.
D
C
A
B
←
→
←
→
2. In this diagram, AB is parallel to CD and the side lengths are as shown. Find the are
Math 444
Geometry for Teachers
Winter 2010
SYLLABUS
Professor:
John M. (Jack) Lee
Padelford C-546, 206-543-1735
[email protected]
Oﬃce hours: to be announced.
TA:
Julie Eaton
Padelford C-404
[email protected]
Discussion sessions: to be ann
Chapter 4
Proofs in Incidence Geometry
In this chapter, we will begin to discuss the process of constructing rigorous mathematical proofs. First, we discuss the general structure of proofs and describe some
“templates” for proofs of different types. Havin
Chapter 3
The Language of Mathematics
In the previous chapter, we introduced incidence geometry, and discussed three of its
four elements as an axiomatic system: undeﬁned terms, axioms, and deﬁnitions. We
have not yet introduced the fourth and most import
Chapter 2
Incidence Geometry
Motivated by the advances described in the previous chapter, mathematicians since
the early twentieth century have always proved theorems using what is now called
the axiomatic method. The purpose of this chapter is to describ
Chapter 1
Reading Euclid
The story of axiomatic geometry begins with Euclid, the most famous mathematician in history. We know very little about Euclid’s life, save that he was a Greek who
lived and worked in Alexandria, Egypt around 300 B . C . E . His m
Appendix D
Conventions for Writing Proofs
Writing mathematical proofs is, in many ways, unlike any other kind of writing.
Over the years, the mathematical community has agreed upon a number of moreor-less standard conventions for proof writing. This appen
Appendix C
Properties of the Real Numbers
Because our axioms for plane geometry are predicated on an understanding of the
real number system, it is important to establish clearly what properties of the real
numbers we are taking for granted. In this appen
Appendix B
Birkhoff’s Axioms for Plane Geometry
These axioms are taken from the 1932 article A set of postulates for plane geometry, based on scale and protractor, by George D. Birkhoff [GDB32]. In Birkhoff’s
system, the undeﬁned terms are point, line, di
Appendix A
Hilbert’s Axioms for Plane Geometry
These axioms are taken from The Foundations of Geometry by David Hilbert (1899),
as translated by E. J. Townsend in 1902 [DH02]. Although Hilbert’s treatment includes axioms for three-dimensional spatial geom
Outline
RHS Perturbations
Pricing Out
Math 407A: Linear Optimization
Lecture 21
Math Dept, University of Washington
November 25, 2009
Lecture 21: Math 407A: Linear Optimization
Math Dept, University of Washington
Outline
RHS Perturbations
Pricing Out
RHS
Outline SILICON CHIP CORPORATION Range Analysis for Objective Coeﬃcients Resource Variations, Marginal Values, and Ran
Math 407A: Linear Optimization
Lecture 20
Math Dept, University of Washington
November 23, 2009
Lecture 20: Math 407A: Linear Optimizati