HW 3 Solutions Math 115, Winter 2009, Prof. Yitzhak Katznelson 7.4 a) Let xn = n2 . Then lim xn = 0 (as you may easily check - for > 0, just let N be 2 ), which is rational, even though all the xn are irrational. b) Consider the number 2 and its decimal e
MATH 4310: CONVERGENCE OF SEQUENCES AND PROOFS
TROY BUTLER
1. A convergent sequence
We should be very comfortable with reproducing the following denition that states precisely what it
means for a sequence of real numbers to converge.
Denition 1. A sequenc
MATH 4310 Spring 2014: Homework #3
Due Thursday February 20
Reading: Chapter 2, Sections 1012
Homework:
Turn all problems in IN THE ORDER THEY ARE ASSIGNED. Each problem needs to be labeled so the
TA can nd your work, and each problem should be neatly pre
MATH 4310 Spring 2014: Homework #2
Due Thursday February 6
Reading: Chapter 2, Sections 79
Homework:
Turn all problems in IN THE ORDER THEY ARE ASSIGNED. Each problem needs to be labeled so the
TA can nd your work, and each problem should be neatly presen
HW #3 Solutions (Math 323)
8.1) a) Let
> 0. Let N = 1 . Then n > N (i.e.,
b) Let
> 0. Let N =
c) Let
> 0. Let N =
1
3
. Then n > N (i.e.,
1
1
n
< ) implies that
1
n1/3
. Then n > N (i.e.,
> 0. Let N = max(6, 4 ). Then n > N (i.e.,
8.2) a) The limit is 0:
MATH 4310 Spring 2014: Homework #1
Due Thursday January 30
Reading: Chapter 1, Sections 15
Homework:
Turn all problems in IN THE ORDER THEY ARE ASSIGNED. Each problem needs to be labeled so the
TA can nd your work, and each problem should be neatly presen
Math 3210-3 HW 13
Solutions
Note: Problems 4 and 7 are extra credit.
Limit Theorems
1. Suppose that lim an = a and lim bn = b. Let sn = using the limit theorems. Proof: We have the following: lim sn = = = = = = lim a3 + 4an n by definition b2 + 1 n lim(a3
Honors Introduction to Analysis I Homework VI
Solution March 23, 2009
Problem 1 Let f be a function dened on a closed domain D. Show that f is continous if and only if the inverse image of every closed set is a closed set. Solution. = Assume f is continuo
MATH 4310: LECTURE 3 - CONTINUITY
TROY BUTLER
1. Connecting ideas and natural extensions of set/sequence concepts
(Go over the basic concept of a function and various notation on the board as these terms come up in the
denitions and theorems.)
Denition 1.
2
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Sequences
Limits of Sequences
A sequence is a function whose domain is a set that has the form
cfw_n Z : n m; m is usually 1 or 0. Thus a sequence is a funct
MATH 4310: LECTURE 2 - PART 2: SERIES
TROY BUTLER
1. Basic Definitions and Notation
(Go over summation notation on board explaining what each symbol in
n
k=m
ak means in the context
of the denition below.)
Denition 1. Given a sequence (ak )n=m of real num
MATH 4310: LECTURE 2 - PART 1: SEQUENCES
TROY BUTLER
1. Basic Definitions and Notation
Aside: I purposefully switch between dierent types of legitimate notational conventions so that you are
comfortable with the variety of correct ways to write things. Yo
MATH 4310: LECTURE 1
TROY BUTLER
1. Basic Notation
Listed below is some of the basic notation used throughout chapter 1 and the rest of the text.
We use curly brackets cfw_ to denote a set of objects.
N is the set of all positive integers given by cfw_1
MATH 4310: WHY DO CAUCHY SEQUENCES CONVERGE AND WHY ARE
CONVERGENT SEQUENCES CAUCHY?
TROY BUTLER
1. What is a Cauchy sequence and why bother with Cauchy sequences?
Many examples of convergent sequences we have looked at like sn = 1/n are simple enough tha
HW 4 Solutions Math 115, Winter 2009, Prof. Yitzhak Katznelson 9.6 Let x1 = 1 and xn+1 = 3x2 for n 1. n a) If a = lim xn , then we have that a = lim xn+1 = lim 3x2 = n 3 lim(xn xn ) = 3(lim xn )2 = 3a2 , where the later steps use various limit laws. Solvi
M413
Introduction to Analysis I
Assignment XIII
Drew Robertson
October 27, 2008
2
+4
Problem 1. Prove lim n +5 = .
n
Discussion 1. M > 0, we want N : n > N
n2 +4
n+5
n2 +4
n+5
> M . That is, we want
n2 +4
n+5
>something. Note
2
> n /6n = n/6. We want n/6
Dynamic Programming
11
Dynamic programming is an optimization approach that transforms a complex problem into a sequence of
simpler problems; its essential characteristic is the multistage nature of the optimization procedure. More so
than the optimizatio