HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!"

Transcription

1 Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following conjectural identity F n x = xn n! For convenience, let us define G n x = xn n! as F n x = G n x. For n 0 and x > 0 we note that G nx = xn n! = xn n! = = = + xn n! x + n =. for n 0 and x > 0, so that can be stated = xn n! n = = G n x. 2 We proceed by induction on n to prove, i.e. F n x = G n x. For n = 0, we immediately verify that G 0 x = ln x = F 0 x. For the inductive step, we assume that is true for n = m. This means that F m x = G m x, while 2 says that G m x = G m+x. Now we calculate F m+ x = 0 F m tdt = 0 G m tdt = lim r G m tdt = lim r G m+tdt = lim G m+ x G m+ r by the Fundamental Theorem of Calculus = G m+ x lim G m+ r In order to complete the inductive step, it remains to show that lim G m+ r = 0. This can be seen by r m writing G m+ r = m +! ln r m+ =. Note that as r goes to 0, the fisrt factor clearly converges to 0 /r whereas the second factor also converges to 0 by L Hopital s rule.

2 Math 7 Fall 205 Now says that n!f n = following inequalities: = = =. Since fx = x n = + n + Hence we have that + ln n ln n =2 n + = n!f n ln n x = =2 n+ x is strictly decreasing for x > 0, we obtain the n x = + = + ln n, x = lnn + > ln n. < ln n =. Since lim + ln n =, we deduce that ln n n ln n n!f n lim =. n ln n Problem A2 Let a, a 2, and b, b 2, be sequences of positive real numbers such that a = b = and b n = b n a n 2 for n = 2, 3,. Assume that the sequence b j is bounded. Prove that converges, and evaluate S. S = a a n n= The relation b n = b n a n 2 for n = 2, 3, can be written as a n = b n + 2 b n. Then for n = 2, 3,, = a a n a a 2 a n = a b b 2 b n b 2 + 2b b n + 2 = 3 b b 2 b n b + 2 b 2 + 2b b n + 2 since a = b = 3b b 2 b n = b + 2b b n + 2 = 3 b b 2 b n b n b + 2b b n + 2 b b 2 b n b n b + 2b b n + 2 = 3 b b 2 b n 2 b + 2b b n + 2 b b 2 b n b n b + 2b b n + 2 b b 2 b n b n For convenience, set c n = for n =, 2,, so that the above identity can be b + 2b b n + 2 written as = 3 a a n 2 c n c n for n = 2, 3,. Then for every positive integer m, we have n= a a 2 a n = a + n=2 a a 2 a n = c n c n = c c m = 3 2 c m, 3 n=2

3 Math 7 Fall 205 where the last identity follows by the fact that c = 3. Note that c m is clearly positive by construction since b, b 2, are all positive. We also now that the sequence b j is bounded by some number M > 0. Now observe that 0 < c m = b b + 2 b 2 b b m b m + 2 = + 2/b + 2/b 2 < + 2/b m + 2/M m. Since + 2/M converges to 3 2. >, this implies that lim m c m = 0. Thus we deduce from 3 that S = n= a a n Problem A3 Let, 2, 3,, 2005, 2006, 2007, 2009, 202, 206, be a sequence defined by x = for =, 2,, 2006 and x + = x + x 2005 for Show that the sequence has 2005 consecutive terms each divisible by Define a sequence y by y = 0 for =, 2,, 2005, y 2006 = and y + = y + y 2005 for Then we can easily compute y 2006 = y 2007 = = y 40 =, y 402 = 2, y 403 = 3,, y 607 = In particular, we have that y 40 = x, y 402 = x 2,, y 607 = x Now an easy strong induction shows that x = y +400 for =, 2,. Note that y is clearly a sequence of integers, so Useful Facts 2 from Lecture 5 see the remar below says that the sequence is periodic modulo In particular, since y, y 2,, y 2005 are all zero and therefore divisible by 2006, there exist infinitely many integers such that y +, y +2,, y are all divisible by Choose such a with 40, then x 4009 = y +, x 4008 = y +2,, x 2005 = y are all divisible by Remar. For the sae of completeness, we state and prove Useful Facts 2 from Lecture 5. Lemma. If x n is a sequence of integers satisfying a linear recurrence x n = a x n +a 2 x n 2 + +a x n, then x n is a periodic sequence in modulo m for every integer m. Proof. Without loss of generality, we may assume that a 0. Consider the vectors of the form x n, x n+,, x n+. Note that there are only finitely many such vectors modulo m, so there exist integers p and N such that x N+p, x N++p,, x N+ +p x N, x N+,, x N+ mod m. 4

4 Math 7 Fall 205 We claim that x n+p x n mod m for all n. It suffices to prove that the following holds for all n: x n+p, x n++p,, x n+ +p x n, x n+,, x n+ mod m. 5 We proceed by the two-way induction on n. The case when n = N is given by 4. For the inductive step, we assume that 5 holds for n = r. We wish to prove that 5 holds for n = r and n = r +. For n = r +, it suffices to prove that x r++p x r+ mod m. This can be seen by x r++p = a x r+ +p + a 2 x r+ 2+p + + a x r+p a x r+ + a 2 x r a x r mod m by recursion by the induction hypothesis = x r+ by recursion For n = r, it suffices to prove that x r +p x r mod m. This can be seen by x r +p = a x r+ +p a x r+ 2+p a x r+p by recursion a x r+ a x r+ 2 a x r mod m by the induction hypothesis = x r by recursion Problem B3 Let x 0 = and for n 0, let x n+ = 3x n + x n 5. In particular, x = 5, x 2 = 26, x 3 = 36, x 4 = 72. Find a closed-form expression for x a means the largest integer a. Note that x n is an integer for n = 0,, 2,. By definition of the floor function, we have the following inequality for n 0: x n 5 < xn 5 xn 5. 6 Then the recursion x n+ = 3x n + x n 5 gives an inequality 3 + 5xn < x n x n. Multiplying by 3 5 yields 4x n 3 5 < 3 5x n+ 4x n. 7

5 Math 7 Fall 205 Now observe that for n =, 2,, x n+ 6x n = 3x n + x n 5 3 5xn < 4x n < 4x n +, x n+ 6x n = 3x n + x n 5 > 3 5xn 4x n where in each line, the first and the second inequality respectively follow from 6 and 7. Hence we have that 4x n < x n+ 6x n < 4x n + for n =, 2,. Since x n+ 6x n is an integer, the only possibility is that x n+ 6x n = 4x n. In other words, we have the following recurrence for n : x n+ = 6x n 4x n. 8 The characteristic polynomial x 2 6x + 4 has two roots 3 ± 5, so there exist constants c and d such that x n = c3 + 5 n + d3 5 n for n = 0,, 2,. Substituiting n = 0 and n =, we obtain a system of equations = c + d and 5 = c d3 5, which can be solved by c = , d = Hence 8 yields the following expression of x 2007 : 0 0 x 2007 =

CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12 CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

More information

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear: MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

More information

Math 2602 Finite and Linear Math Fall 14. Homework 9: Core solutions

Math 2602 Finite and Linear Math Fall 14. Homework 9: Core solutions Math 2602 Finite and Linear Math Fall 14 Homework 9: Core solutions Section 8.2 on page 264 problems 13b, 27a-27b. Section 8.3 on page 275 problems 1b, 8, 10a-10b, 14. Section 8.4 on page 279 problems

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

Lectures 5-6: Taylor Series

Lectures 5-6: Taylor Series Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

More information

I. Pointwise convergence

I. Pointwise convergence MATH 40 - NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously, we have studied sequences of real numbers. Now we discuss the topic of sequences of real valued functions.

More information

CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 Solutions CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

More information

1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution.

1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution. Name : Instructor: Marius Ionescu Instructions: 1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution. 3. If you need extra pages

More information

Introduction. Appendix D Mathematical Induction D1

Introduction. Appendix D Mathematical Induction D1 Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to

More information

Sample Induction Proofs

Sample Induction Proofs Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given

More information

POSITIVE INTEGERS, INTEGERS AND RATIONAL NUMBERS OBTAINED FROM THE AXIOMS OF THE REAL NUMBER SYSTEM

POSITIVE INTEGERS, INTEGERS AND RATIONAL NUMBERS OBTAINED FROM THE AXIOMS OF THE REAL NUMBER SYSTEM MAT 1011 TECHNICAL ENGLISH I 03.11.2016 Dokuz Eylül University Faculty of Science Department of Mathematics Instructor: Engin Mermut Course assistant: Zübeyir Türkoğlu web: http://kisi.deu.edu.tr/engin.mermut/

More information

Math 55: Discrete Mathematics

Math 55: Discrete Mathematics Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

More information

Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012 Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

More information

Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

More information

3.1. Sequences and Their Limits Definition (3.1.1). A sequence of real numbers (or a sequence in R) is a function from N into R.

3.1. Sequences and Their Limits Definition (3.1.1). A sequence of real numbers (or a sequence in R) is a function from N into R. CHAPTER 3 Sequences and Series 3.. Sequences and Their Limits Definition (3..). A sequence of real numbers (or a sequence in R) is a function from N into R. Notation. () The values of X : N! R are denoted

More information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information

Math 115 Spring 2014 Written Homework 3 Due Wednesday, February 19

Math 115 Spring 2014 Written Homework 3 Due Wednesday, February 19 Math 11 Spring 01 Written Homework 3 Due Wednesday, February 19 Instructions: Write complete solutions on separate paper (not spiral bound). If multiple pieces of paper are used, they must be stapled with

More information

Limits at Infinity Limits at Infinity for Polynomials Limits at Infinity for the Exponential Function Function Dominance More on Asymptotes

Limits at Infinity Limits at Infinity for Polynomials Limits at Infinity for the Exponential Function Function Dominance More on Asymptotes Lecture 5 Limits at Infinity and Asymptotes Limits at Infinity Horizontal Asymptotes Limits at Infinity for Polynomials Limit of a Reciprocal Power The End Behavior of a Polynomial Evaluating the Limit

More information

MATH : HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS

MATH : HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS MATH 16300-33: HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS 25-1 Find the absolute value and argument(s) of each of the following. (ii) (3 + 4i) 1 (iv) 7 3 + 4i (ii) Put z = 3 + 4i. From z 1 z = 1, we have

More information

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.

GROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G. Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the

More information

Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 126 Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

More information

x a x 2 (1 + x 2 ) n.

x a x 2 (1 + x 2 ) n. Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

More information

Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

More information

Sequences and Series

Sequences and Series Contents 6 Sequences and Series 6. Sequences and Series 6. Infinite Series 3 6.3 The Binomial Series 6 6.4 Power Series 3 6.5 Maclaurin and Taylor Series 40 Learning outcomes In this Workbook you will

More information

H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

More information

Discrete Mathematics: Homework 7 solution. Due: 2011.6.03

Discrete Mathematics: Homework 7 solution. Due: 2011.6.03 EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that

More information

Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

More information

Math 55: Discrete Mathematics

Math 55: Discrete Mathematics Math 55: Discrete Mathematics UC Berkeley, Spring 2012 Homework # 9, due Wednesday, April 11 8.1.5 How many ways are there to pay a bill of 17 pesos using a currency with coins of values of 1 peso, 2 pesos,

More information

6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, ) 6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

4.3 Limit of a Sequence: Theorems

4.3 Limit of a Sequence: Theorems 4.3. LIMIT OF A SEQUENCE: THEOREMS 5 4.3 Limit of a Sequence: Theorems These theorems fall in two categories. The first category deals with ways to combine sequences. Like numbers, sequences can be added,

More information

INTRODUCTION TO THE CONVERGENCE OF SEQUENCES

INTRODUCTION TO THE CONVERGENCE OF SEQUENCES INTRODUCTION TO THE CONVERGENCE OF SEQUENCES BECKY LYTLE Abstract. In this paper, we discuss the basic ideas involved in sequences and convergence. We start by defining sequences and follow by explaining

More information

16.1. Sequences and Series. Introduction. Prerequisites. Learning Outcomes. Learning Style

16.1. Sequences and Series. Introduction. Prerequisites. Learning Outcomes. Learning Style Sequences and Series 16.1 Introduction In this block we develop the ground work for later blocks on infinite series and on power series. We begin with simple sequences of numbers and with finite series

More information

Math 317 HW #5 Solutions

Math 317 HW #5 Solutions Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show

More information

Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction.

Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction. Course Notes for Math 320: Fundamentals of Mathematics Chapter 3: Induction. February 21, 2006 1 Proof by Induction Definition 1.1. A subset S of the natural numbers is said to be inductive if n S we have

More information

Further linear algebra. Chapter I. Integers.

Further linear algebra. Chapter I. Integers. Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides

More information

Notes: Chapter 2 Section 2.2: Proof by Induction

Notes: Chapter 2 Section 2.2: Proof by Induction Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case - S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example

More information

Mathematical Induction

Mathematical Induction Mathematical Induction MAT30 Discrete Mathematics Fall 016 MAT30 (Discrete Math) Mathematical Induction Fall 016 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)

More information

MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

More information

Section 6-2 Mathematical Induction

Section 6-2 Mathematical Induction 6- Mathematical Induction 457 In calculus, it can be shown that e x k0 x k k! x x x3!! 3!... xn n! where the larger n is, the better the approximation. Problems 6 and 6 refer to this series. Note that

More information

CHAPTER 3. Sequences. 1. Basic Properties

CHAPTER 3. Sequences. 1. Basic Properties CHAPTER 3 Sequences We begin our study of analysis with sequences. There are several reasons for starting here. First, sequences are the simplest way to introduce limits, the central idea of calculus.

More information

Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

More information

Homework # 3 Solutions

Homework # 3 Solutions Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8

More information

Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series

Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series 1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,

More information

Limits and Continuity

Limits and Continuity Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function

More information

1 Limiting distribution for a Markov chain

1 Limiting distribution for a Markov chain Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easy-to-check

More information

MATH PROBLEMS, WITH SOLUTIONS

MATH PROBLEMS, WITH SOLUTIONS MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These

More information

MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform

MATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish

More information

S = k=1 k. ( 1) k+1 N S N S N

S = k=1 k. ( 1) k+1 N S N S N Alternating Series Last time we talked about convergence of series. Our two most important tests, the integral test and the (limit) comparison test, both required that the terms of the series were (eventually)

More information

Lecture 3. Mathematical Induction

Lecture 3. Mathematical Induction Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion

More information

Course Notes for Math 162: Mathematical Statistics Approximation Methods in Statistics

Course Notes for Math 162: Mathematical Statistics Approximation Methods in Statistics Course Notes for Math 16: Mathematical Statistics Approximation Methods in Statistics Adam Merberg and Steven J. Miller August 18, 6 Abstract We introduce some of the approximation methods commonly used

More information

Solutions of Equations in One Variable. Fixed-Point Iteration II

Solutions of Equations in One Variable. Fixed-Point Iteration II Solutions of Equations in One Variable Fixed-Point Iteration II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011

More information

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +... 6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

More information

Probability Generating Functions

Probability Generating Functions page 39 Chapter 3 Probability Generating Functions 3 Preamble: Generating Functions Generating functions are widely used in mathematics, and play an important role in probability theory Consider a sequence

More information

1 Formulating The Low Degree Testing Problem

1 Formulating The Low Degree Testing Problem 6.895 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Linearity Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz In the last lecture, we proved a weak PCP Theorem,

More information

CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5. Number Theory. 1. Integers and Division. Discussion CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

More information

Spring As discussed at the end of the last section, we begin our construction of the rational

Spring As discussed at the end of the last section, we begin our construction of the rational MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 3 The Rational Numbers 1 3.1 The set Q................................. 1 3.2 Addition and multiplication of rational numbers............

More information

More Mathematical Induction. October 27, 2016

More Mathematical Induction. October 27, 2016 More Mathematical Induction October 7, 016 In these slides... Review of ordinary induction. Remark about exponential and polynomial growth. Example a second proof that P(A) = A. Strong induction. Least

More information

Continued fractions and good approximations.

Continued fractions and good approximations. Continued fractions and good approximations We will study how to find good approximations for important real life constants A good approximation must be both accurate and easy to use For instance, our

More information

Inner Product Spaces

Inner Product Spaces Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

More information

k, then n = p2α 1 1 pα k

k, then n = p2α 1 1 pα k Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

More information

MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS

MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 CHAPTER 6. MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 Introduction Recurrence

More information

Solutions series for some non-harmonic motion equations

Solutions series for some non-harmonic motion equations Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 10, 2003, pp 115 122. http://ejde.math.swt.edu or http://ejde.math.unt.edu

More information

sin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2

sin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2 . Problem Show that using an ɛ δ proof. sin() lim = 0 Solution: One can see that the following inequalities are true for values close to zero, both positive and negative. This in turn implies that On the

More information

1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm

1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 005-06-15 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

More information

N E W S A N D L E T T E R S

N E W S A N D L E T T E R S N E W S A N D L E T T E R S 73rd Annual William Lowell Putnam Mathematical Competition Editor s Note: Additional solutions will be printed in the Monthly later in the year. PROBLEMS A1. Let d 1, d,...,

More information

Primitive Prime Divisors of First-Order Polynomial Recurrence Sequences

Primitive Prime Divisors of First-Order Polynomial Recurrence Sequences Primitive Prime Divisors of First-Order Polynomial Recurrence Sequences Brian Rice brice@hmc.edu July 19, 2006 Abstract The question of which terms of a recurrence sequence fail to have primitive prime

More information

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x) SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

More information

On the largest prime factor of x 2 1

On the largest prime factor of x 2 1 On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical

More information

3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.

3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9. SOLUTIONS TO HOMEWORK 2 - MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid

More information

SECTION 10-2 Mathematical Induction

SECTION 10-2 Mathematical Induction 73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

More information

p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6)

p 2 1 (mod 6) Adding 2 to both sides gives p (mod 6) .9. Problems P10 Try small prime numbers first. p p + 6 3 11 5 7 7 51 11 13 Among the primes in this table, only the prime 3 has the property that (p + ) is also a prime. We try to prove that no other

More information

Math 231b Lecture 35. G. Quick

Math 231b Lecture 35. G. Quick Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

More information

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don

More information

MATH 22. THE FUNDAMENTAL THEOREM of ARITHMETIC. Lecture R: 10/30/2003

MATH 22. THE FUNDAMENTAL THEOREM of ARITHMETIC. Lecture R: 10/30/2003 MATH 22 Lecture R: 10/30/2003 THE FUNDAMENTAL THEOREM of ARITHMETIC You must remember this, A kiss is still a kiss, A sigh is just a sigh; The fundamental things apply, As time goes by. Herman Hupfeld

More information

8.7 Mathematical Induction

8.7 Mathematical Induction 8.7. MATHEMATICAL INDUCTION 8-135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture

More information

Mathematical Induction. Lecture 10-11

Mathematical Induction. Lecture 10-11 Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach

More information

Homework until Test #2

Homework until Test #2 MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

More information

Principle of (Weak) Mathematical Induction. P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n))

Principle of (Weak) Mathematical Induction. P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n)) Outline We will cover (over the next few weeks) Mathematical Induction (or Weak Induction) Strong (Mathematical) Induction Constructive Induction Structural Induction Principle of (Weak) Mathematical Induction

More information

CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS

CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in their own right and useful in other areas

More information

Lecture 3: Fourier Series: pointwise and uniform convergence.

Lecture 3: Fourier Series: pointwise and uniform convergence. Lecture 3: Fourier Series: pointwise and uniform convergence. 1. Introduction. At the end of the second lecture we saw that we had for each function f L ([, π]) a Fourier series f a + (a k cos kx + b k

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

COMP 250 Fall Mathematical induction Sept. 26, (n 1) + n = n + (n 1)

COMP 250 Fall Mathematical induction Sept. 26, (n 1) + n = n + (n 1) COMP 50 Fall 016 9 - Mathematical induction Sept 6, 016 You will see many examples in this course and upcoming courses of algorithms for solving various problems It many cases, it will be obvious that

More information

Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)

Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x) ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process

More information

THE 2-ADIC, BINARY AND DECIMAL PERIODS OF 1/3 k APPROACH FULL COMPLEXITY FOR INCREASING k

THE 2-ADIC, BINARY AND DECIMAL PERIODS OF 1/3 k APPROACH FULL COMPLEXITY FOR INCREASING k #A28 INTEGERS 12 (2012) THE 2-ADIC BINARY AND DECIMAL PERIODS OF 1/ k APPROACH FULL COMPLEXITY FOR INCREASING k Josefina López Villa Aecia Sud Cinti Chuquisaca Bolivia josefinapedro@hotmailcom Peter Stoll

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

More information

1. R In this and the next section we are going to study the properties of sequences of real numbers.

1. R In this and the next section we are going to study the properties of sequences of real numbers. +a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

More information

1.3 Induction and Other Proof Techniques

1.3 Induction and Other Proof Techniques 4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

More information

SMT 2014 Algebra Test Solutions February 15, 2014

SMT 2014 Algebra Test Solutions February 15, 2014 1. Alice and Bob are painting a house. If Alice and Bob do not take any breaks, they will finish painting the house in 20 hours. If, however, Bob stops painting once the house is half-finished, then the

More information

Notes on Chapter 1, Section 2 Arithmetic and Divisibility

Notes on Chapter 1, Section 2 Arithmetic and Divisibility Notes on Chapter 1, Section 2 Arithmetic and Divisibility August 16, 2006 1 Arithmetic Properties of the Integers Recall that the set of integers is the set Z = f0; 1; 1; 2; 2; 3; 3; : : :g. The integers

More information

1 Die hard, once and for all

1 Die hard, once and for all ENGG 2440A: Discrete Mathematics for Engineers Lecture 4 The Chinese University of Hong Kong, Fall 2014 6 and 7 October 2014 Number theory is the branch of mathematics that studies properties of the integers.

More information

Taylor Series and Asymptotic Expansions

Taylor Series and Asymptotic Expansions Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.

More information

Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS

Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS Math 507/420 Homework Assignment #1: Due in class on Friday, September 20. SOLUTIONS 1. Show that a nonempty collection A of subsets is an algebra iff 1) for all A, B A, A B A and 2) for all A A, A c A.

More information

Math 317 HW #7 Solutions

Math 317 HW #7 Solutions Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

More information

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5. PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

More information

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b. Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

More information

ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM

ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM Acta Math. Univ. Comenianae Vol. LXXXI, (01), pp. 03 09 03 ON INTEGERS EXPRESSIBLE BY SOME SPECIAL LINEAR FORM A. DUBICKAS and A. NOVIKAS Abstract. Let E(4) be the set of positive integers expressible

More information