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
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 t
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 sequ
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
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
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 =
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
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 =
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 close
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 te
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 be
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
comfortab
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
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 sequence
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 = 3a
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
ECO 202 Midterm Exam
SOLUTIONS
INSTRUCTIONS: Answer all questions in the space provided
TEN Multiple Choice Questions: Circle the correct answer
1/ Malthus predicted population growth would eventually