Math 4443/5443
Quiz (take-home)
Instructions: Prove the following statements. Please indicate which theorems you are
using in your proof.
1. Show that the series
P (x) =
and
x2n
(2n)!
n=0
Q(x) =
x2n+1
(2n + 1)!
n=0
converge pointwise on R and uniformly on
Review for Second Exam
The second exam will cover sections 5.4, 7.3, 8.1, and 8.2 of the text. (The relevant
assignments are assignments 4 through 7.)
The format of the second exam will be similar to that of the rst. Here are the
denitions, statements of
Math 4443/5443
Exam 1
Instructions: Do problems 1, 2 and any ve of the six remaining problems (for a total
of 100 possible points).
1. (20 points)
a. Give the denition of Riemann integral of a function on [a, b].
b. Prove that a function f can have no mor
Math 4443/5443
Exam 2
Instructions: Do problems 1, 2 and any ve of the six remaining problems (for a total
of 100 possible points).
1. (20 points) Give a proof of the following theorem: Suppose f is a Riemann integrable
function on [a, b], and suppose F i
Solutions to Problems on last assignment
9.3.5 (First proof) The given series is of the form
xn yn , where
(yn ) = (1, 1, 1, 1, 1, 1, 1, 1, 1, . . . )
and xn = 1/n. The partial sums of
yn are given by the bounded sequence
(sn ) = (1, 0, 1, 0, 1, 0, 1, 0,
Math 4443/5443
Final Exam
Instructions: Do as many of the problems as you can (total of 100 points).
1. (15 points) Suppose A R, and suppose f : A R, and (fn ) is a sequence of functions
on A. Give denitions of the following statements:
a. f is uniformly
Review for Final Exam
The nal exam will cover all the sections of the text covered on the rst two exams
(5.4, 6.2, 6.4, 7.1, 7.2, 7.3, 8.1, and 8.2), together with sections 8.3, 8.4, 9.1, 9.2, 9.3, and
9.4. It will probably be weighted more towards the ty
Review for First Exam
The rst exam will cover sections 6.2, 6.4, 7.1, and 7.2 of the text. (The relevant
assignments are assignments 1 through 3.)
There will be one or two questions in which I ask you to state a denition or prove
a theorem. Here is a list