Introductory Real Analysis
Math 328, Summer 2015
University of Washington
c 2015, Dr. F. Dos Reis
Homework 2
Due at the beginning of the class on Friday July 17th
Exercise 1. Determine whether the folowing integrals are convergent or divergent
Arctan x
dx
Introductory Real Analysis
Math 327, Spring 2015
1
University of Washington
c
2015,
Dr. F. Dos Reis
Axioms of R
Axiom 1. R is a commutative field.
Definition 1. F + is a commutative field if
1. For any a, b, c in F , a + (b + c) = (a + b) + c, and a (b c)
Introductory Real Analysis II
Math 328, Winter 2016
University of Washington
c
2016,
Dr. F. Dos Reis
Chapter 22
1
Improper integral of the first kind and of the second kind
Definition 1. Improper of the first kind
Let f be an integrable function over ever
Introductory Real Analysis II
Math 328, Winter 2016
University of Washington
c
2016,
Dr. F. Dos Reis
Theory of integration
Definition 1. Given an interval [a, b], a partition of [a, b] is a finite collection of real numbers cfw_xi such that
a = x0 < x1 <
Introductory Real Analysis II
Math 328, Winter 2016
University of Washington
c
2016,
Dr. F. Dos Reis
Continuity, differentiability of functions
1
Limits
Definition 1. Let f be a function defined on a neighborhood of a number a except possibly at a,
lim f
Introductory Real Analysis II
Math 328, Winter 2016
University of Washington
c
2016,
Dr. F. Dos Reis
Chapter 21
1
Radius of convergence
1.1
Definition
Definition 1. A power series is a series of functions in the form f (x) =
X
an xn .
k=0
n
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Introductory Real Analysis
Math 328, Summer 2015
University of Washington
c 2015, Dr. F. Dos Reis
Homework 2
Due at the beginning of the class on Friday July 10th
Exercise 1.
1. Express Arctan x is terms of a power series .
Z x
2. Express
Arctan tdt as a
Mo,Ek^rcjs^ i
(n+Oii
01
,
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rccdl'us
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J
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r
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Introductory Real Analysis
Math 328, Summer 2015
University of Washington
c 2015, Dr. F. Dos Reis
Homework 6
Due at the beginning of the class on Friday August 14th
Exercise 1. Let a and b 2 real numbers such that a < 0 < b. Assume that f is a continuous
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+t
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oor
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is
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Introductory Real Analysis
Math 328, Summer 2015
University of Washington
c 2015, Dr. F. Dos Reis
Homework 5
Due at the beginning of the class on Friday August 7tht
Exercise 1. Given the function (x) =
1
1
dt.
1 + tx
1. For which values of x is convergent
Introductory Real Analysis
Math 328, Summer 2015
University of Washington
c 2015, Dr. F. Dos Reis
Homework 4
Due at the beginning of the class on Friday July 31st
Exercise 1. The gaol of the exercise is to study the Wallis integrals and prove that
24n (p!
Introductory Real Analysis
Math 328, Summer 2015
University of Washington
c 2015, Dr. F. Dos Reis
Homework 2
Due at the beginning of the class on Friday July 10th
Exercise 1.
1. Express Arctan x is terms of a power series .
x
Arctan tdt as a power series
Introductory Real Analysis
Math 328, Summer 2015
University of Washington
c
2015,
Dr. F. Dos Reis
Last Name (PRINT):
First Name (PRINT):
Summer 2015 Introductory Real Analysis II
Second Examination
Instructions
1. The use of all electronic devices and any
Introductory Real Analysis
Math 327, Summer 2015
University of Washington
c 2015, Dr. F. Dos Reis
Last Name (PRINT):
First Name (PRINT):
Summer 2015 Introductory Real Analysis
Second Examination
Instructions
1. The use of all electronic devices is prohibi