MATH 425a
SAMPLE MIDTERM EXAM 1
Fall 2011
Prof. Alexander
(1)(a) State what it means for a point p to be a limit point of a set E .
(b) State the converse of the following (dont worry about whether th
MATH 425a ASSIGNMENT 1
FALL 2011 Prof. Alexander
Due Friday September 2.
Rudin Chapter 1 #1, 2, 4, 5 plus the problems (I) - (V) below:
(I) Take the set of all real numbers (not complex!) as the unive
MATH 425a
SAMPLE MIDTERM EXAM 1 SOLUTIONS
Fall 2011
Prof. Alexander
(1)(a) Every neighborhood of p contains a point x E with x = p.
(b) If x is not in the interior of E c then x is a limit point of E
MATH 425a SAMPLE FINAL EXAM SOLUTIONS
FALL 2011 Prof. Alexander
(1) Let > 0. There exists N such that n N = |fn (t) f (t)| for all t [a, b]. Then
for all x [a, b] we have
x
x
fn (t) dt
n N = |Fn (x)
MATH 425a SAMPLE MIDTERM 2 SOLUTIONS
FALL 2011 Prof. Alexander
(1)(a)
an converges if the partial sums sn =
(b) See text.
n
j =1
form a converging sequence.
(2)(a)
(1)n
1.5n
n1.5
1/n
=
1.5
(n1/n )1.5
MATH 425a
SAMPLE MIDTERM EXAM 2
Fall 2011
Prof. Alexander
This is probably a little too long to realistically complete in an hour (especially with
#5a), but the problems are typical of what could be o
MATH 425a ASSIGNMENT 10
FALL 2011 Prof. Alexander
Due Wed. December 7, noon.
Rudin Chapter 7 #1, 2, plus the problems (A)(C) below, are to be turned in. Chapter 2
#3, 4, 8, 9 are for practice for the
MATH 425a ASSIGNMENT 10
FALL 2011 Prof. Alexander
Due Monday November 28.
Rudin Chapter 5 #26, Chapter 6 #1, 2, 3ab, 8, plus the problems (I)-(V) below. Problems
1, 2, and (V) should be relatively qui
MATH 425a ASSIGNMENT 9
FALL 2011 Prof. Alexander
Due Wednesday November 16.
Rudin Chapter 5 #1, 2, 6, 7, 12, 13ab plus the problems (A)(E) below:
(A) A function f on R is called even if f (x) = f (x)
MATH 425a ASSIGNMENT 8
FALL 2011 Prof. Alexander
Due Monday November 7.
The due date is after the midterm, but this material IS covered on the midterm.
Rudin Chapter 4 #11 (rst sentence only), 12, 18,
MATH 425a ASSIGNMENT 7
FALL 2011 Prof. Alexander
Due Wednesday October 26.
Rudin Chapter 3 #11bc, Chapter 4 #2, 4, plus the problems (I)(VI) below:
(I) If
an converges, show that
Test wont work.)
an
n
MATH 425a ASSIGNMENT 6
FALL 2011 Prof. Alexander
Due Wednesday October 19.
Rudin Chapter 3 #7, 10, 20, 21 plus the problems (A)(F) below:
(A) Suppose an > 0 and
n
an converges. Show that
n
a2 converge
MATH 425a ASSIGNMENT 5
FALL 2011 Prof. Alexander
Due Wednesday October 12.
Rudin Chapter 2 #19, Chapter 3 #1, 3, 5, plus the problems (I)(VIII) below:
(I) Suppose sn 2, and tn 3 for all n.
(a) If snk
MATH 425a ASSIGNMENT 4
FALL 2011 Prof. Alexander
Due Monday October 3.
Note this assignment is due after Midterm 1, but the material IS covered on the midterm.
Rudin Chapter 2 #12, 14, 16, 22 plus the
MATH 425a ASSIGNMENT 3
FALL 2011 Prof. Alexander
Due Wednesday September 21.
Rudin Chapter 2 #5, 7b, 8, 9ac, 11 (omit d5 ) plus the problems (I) - (V) below:
(I) Suppose A1 , . . . An are subsets of a
MATH 425a ASSIGNMENT 2
FALL 2011 Prof. Alexander
Due Monday September 12.
Normally this would be due on a Friday, but it is Monday this time since Im out of town.
Rudin Chapter 1 #12, 13, 17, Chapter