Math 4100/6100
Fall 2006
Final Exam
Math 4100 students: Answer questions 18 Math 6100 students: Answer all of the questions below
1. (15 points) Give examples of the following. No proofs are required. (a) A function that is continuous on a bounded set but
Math 4100/6100, Midterm exam study sheet
1. Prove from rst principles that
3n2 + 3
=3
n n2 + 2
lim
2. Let S be a bounded subset of R. Prove that there exists a sequence (xn )n such that
all xn S
limn xn = sup S
3. (Equivalence of the two denitions of fu
Math 4100/6100, HW #5
Due Friday, November 15
1. Suppose f is dened on all of R, and satises
|f (x) f (y )| (x y )2
for all x, y R. Prove that f is constant.
2. Suppose f is dierentiable on all of R, and that limx f (x) = 0. Show that
lim (f (x + 1) f (x)
Math 4100/6100, HW #4
Due Wednesday, October 23
1. A denition: Given a set S R and a function f (x) dened only for x S (i.e. the
domain of f is S ), let us say that limxa f (x) = L if, for every > 0 there is > 0 such
that
x S, 0 < |x a| < = |f (x) L| <
a)
Math 4100/6100, HW #3
Due Monday, September 23
1. Compute the radius of convergence of each of the following power series.
xn
a) 2n+1
n=1
2n+1
b) x n
n=1
xn
c) n!
n=1
2. By manipulating the power series
cos x = 1
sin x = x
x2 x4
+
=
2
24
x3
6
+
x5
120
Math 4100/6100, HW #2
Due Wednesday, September 4
1. Dene the real number c := supcfw_x R : x2 < 10. Prove that c2 = 10.
2. Let (xn )n be a sequence of positive real numbers that has no convergent subsequence.
Prove that limn xn = +. What if the xn are per
Math 4100/6100, HW #1
1. Let A, B be two bounded subsets of the real line R. Dene new subsets of R by
A + B := cfw_a + b : a A, b B := cfw_c : there exist a A, b B such that c = a + b
A B := cfw_a b : a A, b B
Prove or give a counterexample:
a) sup(A +
Math 4100/6100, HW #1 solutions
1. Let A, B be two bounded subsets of the real line R. Dene new subsets of R by
A + B := cfw_a + b : a A, b B := cfw_c : there exist a A, b B such that c = a + b
A B := cfw_a b : a A, b B
Prove or give a counterexample:
a
MATH] 41006100 {Azoffl Ealll 2011
Study Aids for Eirst Hour Exant
The problems on this page provide practice with relatively straightforward ark
plications of the denitionsl
Thd next page reproduces an exam given some years ago and indicated mediar
sca
Math 4100/6100, Fall 2013
instructor J. Fu
course webpage http:/www.math.uga.edu/fu/math4100.htm
email fu@math.uga.edu
oce 407 Boyd Grad Studies
phone 2-2562
oce hours MW 1:302:30 pm, T 3:304:30, or by appointment
grader Kenny Jacobs, kjacobs@math.uga.edu