Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Homework Assignment #8
Homework due:
Tuesday 5/24/11 at beginning of discussion section
Reading material. Read sections 2.11, 2.12, 2.13 in the textbook.
Problems
1. For each of the following sequen
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Solutions to Homework Assignment #7
1. Prove that the sequence
an =
1 3 5 (2n 1)
2 4 6 (2n)
converges.
Proof. We will apply the monotone convergence theorem. Note that since 2n1 < 1 we have that an+
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Homework Assignment #7
Homework due:
Tuesday 5/17/11 at beginning of discussion section
Reading material. Read sections 2.9, 2.10 in the textbook.
Problems
1. Prove that the sequence
an =
1 3 5 . .
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Solutions to Homework Assignment #6
1. Let sequences (sn ) and (tn ) converge to respective limits S and
n=1
n=1
T . Fix some > 0. By denition, there exist numbers ms , mt such
that for any ns ms an
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Homework Assignment #6
Homework due:
Tuesday 5/10/11 at beginning of discussion section
Reading material. Read section 2.8 in the textbook.
Problems
1. Let (sn ) and (tn ) be sequences that converge
Math 25: Advanced Calculus
UC Davis, Spring 2011
Solutions to homework assignment #5
Reading material. Read sections 2.42.7 in the textbook.
Problems on previous material
1. Answer the following problems in the textbook: A.7.1, A.7.3, A.8.2,
A.8.5, 1.9.2,
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Homework Assignment #5
Note. This homework will not be collected or graded. It is meant
as a set of practice problems to aid in studying for the midterm exam.
Solutions for selected problems will be
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Solutions to Homework Assignment #4
1. Using the Archimedean Theorem, prove each of the three statements that follow the proof of that theorem
in section 1.7 of the textbook.
(a) No matter how large
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Homework Assignment #4
Homework due:
Tuesday 4/26/11 at beginning of discussion section
Reading material. Read sections 1.7, 1.9, 1.10, 2.2, 2.4 in the textbook.
Problems
1. Using the Archimedean Th
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Solutions to Homework Assignment #3
Solutions
1. Let F be a eld.
(a) Fix some a, b, c F . By commutativity of multiplication (axiom A1), distributivity of multiplication over addition (axiom
AM1), a
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Homework Assignment #3
Homework due:
Tuesday 4/19/11 at beginning of discussion section
Reading material. Read sections 1.11.6 in the textbook.
Problems
1. Let F be a eld (a set with operations + an
Spring 2011
Solutions to Homework 2
MAT 25
1. Prove by induction that for every n N
12 + 22 + + n2 =
n(n + 1)(2n + 1)
.
6
Proof. When n = 1, the statement becomes
1 = 12 =
1(1 + 1)(2 + 1)
6
= = 1,
6
6
which is clearly true.
Now, assume that
12 + 22 + + n2
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Solutions to Homework Assignment #8
1. (a) lim sup an = 1; lim inf an = 1; the set of subsequential limits is
n
n
cfw_1, 1.
(b) lim sup an = 1; lim inf an = 1; the set of subsequential limits is
n
1
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Homework Assignment #9
Homework due:
Tuesday 5/31/11 at beginning of discussion section
Reading material.
book.
Read section 3.4, 3.5, 3.6.1, 3.6.2, 3.6.12 in the text-
Problems
1. Decide whether or
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Solutions to Homework Assignment #9
1. Decide whether or not each of the following innite series converges or diverges. Prove your claims.
(a) The series
n=1
1
nn
diverges.
n
Proof. Consider the seq
201A Midterm - November 1, 2010 - Thomases
Name:
Show all of your work, in particular note any theorems you may use and
state the conditions carefully and make sure they are satised.
Problem Possible Points Points Received
1
10
2
10
3
10
4
10
5
10
Total
5
201A, Fall 10, Thomases
Homework 5 Solutions
Mihaela Ifrim
1. If f is continuous on [0, 1] and if
1
f (x)xn dx = 0,
n N ,
0
then f (x) = 0 on [0, 1].
Hint. The integral of the product of f with any polynomial is zero.
1
Use the Weierstrass theorem to show
201A, Fall 10, Thomases
Homework 4 - Solutions
Mihaela Ifrim
1. Suppose fn C ([0, 1]) is a monotone decreasing sequence that converges pointwise to f C ([0, 1]). Prove that fn converges uniformly
to f . This result is called Dini s monotone convergence th
201A, Fall 10, Thomases
Homework 3
Mihaela Ifrim
1. Let X be a normed linear space. A series in
xn in X is absolutely
convergent if
xn converges to a nite value in R. Prove X is a
Banach space if and only if every absolutely convergent series converges.
P
201A, Fall 10, Thomases
Homework 2
1. Show that if f is a function, and S is a bounded subset of R, the the
following statements are equivalent:
(i) The function f is uniformly continuous on S .
(ii) If cfw_xn is a Cauchy sequence in S then cfw_f (xn ) i
MAT201A - HOMEWORK 1 SOLUTIONS
MIHAELA IFRIM
1. Let (X, d) be a metric space. Suppose that cfw_xn X is a sequence and set n := d(xn , xn+1 ).
Show that for m > n
m1
d(xn , xm )
k
k=n
k .
k=n
Conclude from this that if
d(xn , xn+1 ) <
k =
k=1
n=1
then
Linear Algebra
As an Introduction to Abstract Mathematics
Lecture Notes for MAT67
University of California, Davis
written Fall 2007, last updated February 13, 2010
Isaiah Lankham
Bruno Nachtergaele
Anne Schilling
Copyright c 2007 by the authors.
These lec
Linear Algebra
As an Introduction to Abstract Mathematics
Lecture Notes for MAT67
University of California, Davis
written Fall 2007, last updated September 24, 2011
Isaiah Lankham
Bruno Nachtergaele
Anne Schilling
Copyright c 2007 by the authors.
These le
Homework Assignment #1
Homework due.
Math 67
UC Davis, Fall 2011
Tuesday 10/4/11 at discussion section.
Reading material. Read Chapters 12 and Appendix A in the textbook.
Problems
1. For each of the following systems of linear equations, if the system has
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Solutions to practice problems
Question 1
n
For n = 0, 1, 2, 3, . . . and 0 k n dene numbers Ck by
n
Ck =
n!
n(n 1) . . . (n k + 1)
=
k !(n k )!
k!
n
n
(for k = 0 and k = n we dene C0 = Cn = 1).
(a)
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Practice problems for the nal
Important notes
The nal will be 2 hours long.
The nal will be a closed-book exam.
All of the course material will be covered (except parts that
were explicitly descr
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Homework Assignment #2
Homework due:
Tuesday 4/12/11 at beginning of discussion section
Homework guidelines. 1 As you may have discovered in last weeks
assignment, writing a good proof is not easy.
Math 25: Advanced Calculus
UC Davis, Spring 2011
Math 25 Solutions to Homework Assignment #1
Remember throughout that there may be more than one correct way to prove
any given result.
Solutions
1. Let n be a natural number, divisible both by 5 and by 3. B