# 10.2 Series and Convergence

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and determine the convergence or divergence of the series Use geometric series to model and solve real-life problems

2 Sigma Notation The (finite) sum a 1 + a 2 + a a N can be written as N i=1 i is called the index of summation and 1 and N are the lower and upper limits of summation, respectively. a i

3 Examples 1 + 1/2 + 1/3 + 1/4 + 1/5 = 5 i=1 1 i N ( 1)N 16 2 N = ( 1) i 1 2 i. i=0 4(1) + 4 ( ) ( ) ( ) = i=1 ( ) 1 4 i 2

4 Infinite Series and Partial Sums The infinite summation a n = a 1 + a 2 + a is called an infinite series. The sequence of partial sums of the series is denoted by S 1 = a 1, S 2 = a 1 +a 2, S 3 = a 1 +a 2 +a 3,..., S n = a 1 +a a n.

5 Convergence and divergence of an infinite series We say that an infinite series converges to (a single finite value) S if the sequence of partial sums {S n } converges to S. We write lim S n = n a n = S. Here S is called the sum of the series. If the limit of the sequence of partial sums {S n } does not exist, then the series diverges.

6 Example: Determining convergence or divergence Consider the infinite series To see if the series converges or diverges we look at the sequence of partial sums: S 1 = n S 2 = = 3 4 S 3 = = 7 8 S 4 = = 15 16

7 Example: cont. Consider the infinite series 1 2 n We are starting to see a pattern? In general S n = 2n 1 2 n. So to determine the sum of the infinite series, now all we have to figure out is what does {S n } converge to (if it does). But we have seen this before and we know that lim n S n = 1, so the infinite series sums to 1.

8 The n th term test for divergence One easy way to see if a series diverges is to check if the individual terms of the series are going to zero, if they are not then the series has to diverge. n th -Term test for divergence Consider the series a n. If then the series diverges. lim a n 0 n

9 Example Determine if the following series converge: 2 n 2 n If we look at we see that as lim n 2 n 2 n n 2n 2 n = 2 n+1 2 n = 1/ n+1 2 n+1 2 n+1 So we see that as n the terms a n 1 2 must diverge. 0 so the series

10 Geometric Series A series of the form where r is any real number is called a geometric series. It has special known convergence properties. In particular if r < 1 then this series converges: r n r n = 1 1 r if r < 1.

11 Geometric series example If r = 1 2, since r < 1 we have that ( ) 1 n = = 2. How is this different from our previous example 1 2 n = 1. Note that the second series starts at n = 1, but So these two are in agreement. 1 2 n = n = = 2

12 Geometric Series cont. If r 1 then the geometric series will diverge. As an example consider r = 2 2 n, has partial sums S 0 = 1 S 1 = = 3 S 2 = = 7 S 3 = = 15 So we see that S n meaning that the sum diverges.

13 Example Find the sum of the following series: 4 3 n+1 Since this isn t quite in the right form to apply the formula we can factor out both the 4 and one copy of 1 3 to get 4 3 n+1 = 4 ( ) ( ) 1 n = = = 2 3

14 Modeling a bouncing ball Suppose a ball is dropped from a height of 6 feet and begins to bounce. The height of each bounce is 3 4 that of the preceding bounce. Find the total distance travelled by the ball. Solution: When the ball hits the ground the first time it has travelled D 1 = 6 feet. Between the first and second bounces the ball will travel D 2 = 3 4 (6) (6) = 3 4 (12) feet. Between the second and third bounces the ball will travel D 3 = ( 3 2 4) (12) and so on, so between the (n 1) th and n th bounces the ball will travel D n = ( 3 n 1 4) (12).

15 Modeling a bouncing ball Suppose a ball is dropped from a height of 6 feet and begins to bounce. The height of each bounce is 3 4 that of the preceding bounce. Find the total distance travelled by the ball. Solution: When the ball hits the ground the first time it has travelled D 1 = 6 feet. Between the first and second bounces the ball will travel D 2 = 3 4 (6) (6) = 3 4 (12) feet. Between the second and third bounces the ball will travel D 3 = ( 3 2 4) (12) and so on, so between the (n 1) th and n th bounces the ball will travel D n = ( 3 n 1 4) (12).

16 bouncing ball cont. Summing these distances (after D 1 ) we have ( ) 3 n distance = 6 + (12) 4 Now this is almost the geometric series, we know ( ) 3 n (12) = /4 = 48 and we can see that ( ) 3 n ( ) 3 n (12) = 12 + = Therefore the sum we want is 36. Hence ( ) 3 n distance = 6 + (12) = = 42 feet 4

### Math Example Set 14A Section 10.1 Sequences Section 10.2 Summing an Infinite Series

Math 10360 Example Set 14A Section 10.1 Sequences Introduction to Geometric Sequence and Series 1. A ball is projected from the ground to a height of 10 feet and allowed to freely fall and rebound when

### Section 1.3 P 1 = 1 2. = 1 4 2 8. P n = 1 P 3 = Continuing in this fashion, it should seem reasonable that, for any n = 1, 2, 3,..., = 1 2 4.

Difference Equations to Differential Equations Section. The Sum of a Sequence This section considers the problem of adding together the terms of a sequence. Of course, this is a problem only if more than

### EXAM. Practice for Third Exam. Math , Fall Dec 1, 2003 ANSWERS

EXAM Practice for Third Exam Math 35-006, Fall 003 Dec, 003 ANSWERS i Problem series.) A.. In each part determine if the series is convergent or divergent. If it is convergent find the sum. (These are

### To discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes.

INFINITE SERIES SERIES AND PARTIAL SUMS What if we wanted to sum up the terms of this sequence, how many terms would I have to use? 1, 2, 3,... 10,...? Well, we could start creating sums of a finite number

### 2 n = n=1 a n is convergent and we let. i=1

Lecture 4 : Series So far our definition of a sum of numbers applies only to adding a finite set of numbers. We can extend this to a definition of a sum of an infinite set of numbers in much the same way

### GEOMETRIC SEQUENCES AND SERIES

. Geometric Sequences and Series (-7) 63 a novel and every day thereafter increase their daily reading by two pages. If his students follow this suggestion, then how many pages will they read during October?,085

### 4/1/2017. PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY

PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY 1 Oh the things you should learn How to recognize and write arithmetic sequences

### Project: The Harmonic Series, the Integral Test

. Let s start with the sequence a n = n. Does this sequence converge or diverge? Explain. To determine if a sequence converges, we just take a limit: lim n n sequence converges to 0. = 0. The. Now consider

### 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

12.3 Geometric Sequences Copyright Cengage Learning. All rights reserved. 1 Objectives Geometric Sequences Partial Sums of Geometric Sequences What Is an Infinite Series? Infinite Geometric Series 2 Geometric

### 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

### Arithmetic Progression

Worksheet 3.6 Arithmetic and Geometric Progressions Section 1 Arithmetic Progression An arithmetic progression is a list of numbers where the difference between successive numbers is constant. The terms

### GEOMETRIC SEQUENCES AND SERIES

4.4 Geometric Sequences and Series (4 7) 757 of a novel and every day thereafter increase their daily reading by two pages. If his students follow this suggestion, then how many pages will they read during

### AFM Ch.12 - Practice Test

AFM Ch.2 - Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question.. Form a sequence that has two arithmetic means between 3 and 89. a. 3, 33, 43, 89

### Objectives. Materials

Activity 5 Exploring Infinite Series Objectives Identify a geometric series Determine convergence and sum of geometric series Identify a series that satisfies the alternating series test Use a graphing

### 2 1 2 s 3 = = s 4 = =

Infinite Series - EPM c CNMiKnO PG - The word series as used in mathematics is different from the way it is used in everyday speech. Webster s Third New International Dictionary includes the following

### 8.3. GEOMETRIC SEQUENCES AND SERIES

8.3. GEOMETRIC SEQUENCES AND SERIES What You Should Learn Recognize, write, and find the nth terms of geometric sequences. Find the sum of a finite geometric sequence. Find the sum of an infinite geometric

### 10-3 Geometric Sequences and Series

1 2 Determine the common ratio, and find the next three terms of each geometric sequence = 2 = 2 The common ratio is 2 Multiply the third term by 2 to find the fourth term, and so on 1( 2) = 2 2( 2) =

### JANUARY TERM 2012 FUN and GAMES WITH DISCRETE MATHEMATICS Module #9 (Geometric Series and Probability)

JANUARY TERM 2012 FUN and GAMES WITH DISCRETE MATHEMATICS Module #9 (Geometric Series and Probability) Author: Daniel Cooney Reviewer: Rachel Zax Last modified: January 4, 2012 Reading from Meyer, Mathematics

### Geometric Series. A Geometric Series is an infinite sum whose terms follow a geometric progression that is, each pair of terms is in the same ratio.

Introduction Geometric Series A Geometric Series is an infinite sum whose terms follow a geometric progression that is, each pair of terms is in the same ratio. A simple example wound be: where the first

### A power series about x = a is the series of the form

POWER SERIES AND THE USES OF POWER SERIES Elizabeth Wood Now we are finally going to start working with a topic that uses all of the information from the previous topics. The topic that we are going to

### Properties of sequences Since a sequence is a special kind of function it has analogous properties to functions:

Sequences and Series A sequence is a special kind of function whose domain is N - the set of natural numbers. The range of a sequence is the collection of terms that make up the sequence. Just as the word

### 10.3 GEOMETRIC AND HARMONIC SERIES

10.3 Geometric and Harmonic Series Contemporary Calculus 1 10.3 GEOMETRIC AND HARMONIC SERIES This section uses ideas from Section 10.2 about series and their convergence to investigate some special types

### Section 3 Sequences and Limits

Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.

### Lab 12: Introduction to Series

Lab 2: Introduction to Series The purpose of this laboratory experience is to develop fundamental understanding of series and their convergence or divergence. We will study basic types of series such as

### 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

### Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

### Grade 7/8 Math Circles. Sequences

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles October 18 th /19 th Sequences A Sequence of Mathematical Terms What is a sequence?

### 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

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### Convergence of Power Series Lecture Notes

Convergence of Power Series Lecture Notes Consider a power series, say \$ % 0aB œ B B B B â. Does this series converge? This is a question that we have een ignoring, ut it is time to face it. Whether or

### Lecture 24: Series finale

Lecture 24: Series finale Nathan Pflueger 2 November 20 Introduction This lecture will discuss one final convergence test and error bound for series, which concerns alternating series. A general summary

### #1-12: Write the first 4 terms of the sequence. (Assume n begins with 1.)

Section 9.1: Sequences #1-12: Write the first 4 terms of the sequence. (Assume n begins with 1.) 1) a n = 3n a 1 = 3*1 = 3 a 2 = 3*2 = 6 a 3 = 3*3 = 9 a 4 = 3*4 = 12 3) a n = 3n 5 Answer: 3,6,9,12 a 1

### I remember that when I

8. Airthmetic and Geometric Sequences 45 8. ARITHMETIC AND GEOMETRIC SEQUENCES Whenever you tell me that mathematics is just a human invention like the game of chess I would like to believe you. But I

### Math Circles April 1st, 2015 Infinite Series Instructor: Jordan Hamilton NOTES

Math Circles April st, 05 Infinite Series Instructor: Jordan Hamilton NOTES A Card Trick Consider a standard deck of 5 cards stacked at the edge of a table If you push the first one off the stack, how

### 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)

### Geometric Series and Annuities

Geometric Series and Annuities Our goal here is to calculate annuities. For example, how much money do you need to have saved for retirement so that you can withdraw a fixed amount of money each year for

### The Limit of a Sequence of Numbers: Infinite Series

Connexions module: m36135 1 The Limit of a Sequence of Numbers: Infinite Series Lawrence Baggett This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License

### Calculus for Middle School Teachers. Problems and Notes for MTHT 466

Calculus for Middle School Teachers Problems and Notes for MTHT 466 Bonnie Saunders Fall 2010 1 I Infinity Week 1 How big is Infinity? Problem of the Week: The Chess Board Problem There once was a humble

### 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

### The Alternating Series Test

The Alternating Series Test So far we have considered mostly series all of whose terms are positive. If the signs of the terms alternate, then testing convergence is a much simpler matter. On the other

### Math 115 Spring 2011 Written Homework 5 Solutions

. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence

### The Geometric Series

The Geometric Series Professor Jeff Stuart Pacific Lutheran University c 8 The Geometric Series Finite Number of Summands The geometric series is a sum in which each summand is obtained as a common multiple

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### 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

### 1. Sequences as Functions

Geometric Sequences and Series p.. Sequences as Functions A sequence can be thought of as a function, with the input numbers consisting of the natural numbers, and the output numbers being the terms. In

### Overview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series

Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them

### Sequences with MS

Sequences 2008-2014 with MS 1. [4 marks] Find the value of k if. 2a. [4 marks] The sum of the first 16 terms of an arithmetic sequence is 212 and the fifth term is 8. Find the first term and the common

### SEQUENCES AND SERIES. For example: 2, 5, 8, 11,... is an arithmetic sequence with d = 3.

SEQUENCES AND SERIES A sequence is a set of numbers in a defined order (a pattern). There are two kinds of sequences: arithmetic and geometric. An arithmetic sequence uses a number called d (difference)

### Examples of infinite sample spaces. Math 425 Introduction to Probability Lecture 12. Example of coin tosses. Axiom 3 Strong form

Infinite Discrete Sample Spaces s of infinite sample spaces Math 425 Introduction to Probability Lecture 2 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 4,

### 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

### 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.

### M11PC - UNIT REVIEW: Series & Sequences

Name: Class: Date: ID: A M11PC - UNIT REVIEW: Series & Sequences Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The first three terms of the sequence

### Convergent Sequence. Definition A sequence {p n } in a metric space (X, d) is said to converge if there is a point p X with the following property:

Convergent Sequence Definition A sequence {p n } in a metric space (X, d) is said to converge if there is a point p X with the following property: ( ɛ > 0)( N)( n > N)d(p n, p) < ɛ In this case we also

### 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

### Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

### Lecture 2 : Basics of Probability Theory

Lecture 2 : Basics of Probability Theory When an experiment is performed, the realization of the experiment is an outcome in the sample space. If the experiment is performed a number of times, different

### (a) Prove that the sum of the first n terms of this series is given by. 10 k 100(2 ).

1. A geometric series is a + ar + ar 2 +... (a) Prove that the sum of the first n terms of this series is given by (b) Find S n n a(1 r ) =. 1 r (4) (c) Find the sum to infinity of the geometric series

### 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

### Aritmetic and Geometric Sequences Review Problems

Aritmetic and Geometric Sequences Review Problems Mr. Konstantinou 14/5/2013 1 1. The sixth term of an arithmetic sequence is 24. The common difference is 8. Calculate the first term of the sequence. The

### 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

### 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

### About the Gamma Function

About the Gamma Function Notes for Honors Calculus II, Originally Prepared in Spring 995 Basic Facts about the Gamma Function The Gamma function is defined by the improper integral Γ) = The integral is

### 2 Sequences and Accumulation Points

2 Sequences and Accumulation Points 2.1 Convergent Sequences Formally, a sequence of real numbers is a function ϕ : N R. For instance the function ϕ(n) = 1 n 2 for all n N defines a sequence. It is customary,

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### 6.8 Taylor and Maclaurin s Series

6.8. TAYLOR AND MACLAURIN S SERIES 357 6.8 Taylor and Maclaurin s Series 6.8.1 Introduction The previous section showed us how to find the series representation of some functions by using the series representation

### Sucesiones repaso Klasse 12 [86 marks]

Sucesiones repaso Klasse [8 marks] a. Consider an infinite geometric sequence with u = 40 and r =. (i) Find. u 4 (ii) Find the sum of the infinite sequence. (i) correct approach () e.g. u 4 = (40) u 4

### Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 3-7, 3-8 The remaining discrete random

### Mathematical Methods of Engineering Analysis

Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................

### 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

### 61. REARRANGEMENTS 119

61. REARRANGEMENTS 119 61. Rearrangements Here the difference between conditionally and absolutely convergent series is further refined through the concept of rearrangement. Definition 15. (Rearrangement)

### m i: is the mass of each particle

Center of Mass (CM): The center of mass is a point which locates the resultant mass of a system of particles or body. It can be within the object (like a human standing straight) or outside the object

### Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover

### Activity 5a Potential and Kinetic Energy PHYS 010. To investigate the relationship between potential energy and kinetic energy.

Name: Date: Partners: Purpose: To investigate the relationship between potential energy and kinetic energy. Materials: 1. Super-balls, or hard bouncy rubber balls. Metre stick and tape 3. calculator 4.

### MATH 105: PRACTICE PROBLEMS FOR SERIES: SPRING Problems

MATH 05: PRACTICE PROBLEMS FOR SERIES: SPRING 20 INSTRUCTOR: STEVEN MILLER (SJM@WILLIAMS.EDU).. Sequences and Series.. Problems Question : Let a n =. Does the series +n+n 2 n= a n converge or diverge?

### MATH 2300 review problems for Exam 3 ANSWERS

MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test

### Supporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Algebra and coordinate geometry: Module 11. Sequences and series

1 Supporting Australian Mathematics Project 3 4 5 6 7 8 9 10 11 1 A guide for teachers Years 11 and 1 Algebra and coordinate geometry: Module 11 Sequences and series Sequences and series A guide for teachers

### GEOMETRIC SEQUENCES AND SERIES

GEOMETRIC SEQUENCES AND SERIES Summary 1. Geometric sequences... 2 2. Exercise... 6 3. Geometric sequence applications to financial mathematics... 6 4. Vocabulary... 7 5. Exercises :... 8 6. Geometric

### 15.1. Integration as the limit of a sum. Introduction. Prerequisites. Learning Outcomes. Learning Style

Integration as the limit of a sum 15.1 Introduction In Chapter 14, integration was introduced as the reverse of differentiation. A more rigorous treatment would show that integration is a process of adding

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### 7.1 Convergence of series: integral test and alternating series

7 Infinite series This topic is addressed in Chapter 10 of the textbook, which has ten sections and gets about four weeks in Math 104. That s because it is an important foundation for differential equations,

### 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

### Sequence of Numbers. Mun Chou, Fong QED Education Scientific Malaysia

Sequence of Numbers Mun Chou, Fong QED Education Scientific Malaysia LEVEL High school after students have learned sequence. OBJECTIVES To review sequences and generate sequences using scientific calculator.

### 1 Norms and Vector Spaces

008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

### Lab 10: The Alternating Harmonic Series

Lab 10: The Alternating Harmonic Series Names: Introduction In this lab we will explore the alternating harmonic series: 1-1 2 + 1 3-1 4 + 1 5 - The terms in this series clearly approach zero, but we know

### Algebra 2 - Arithmetic Sequence Practice Date: Period:

Algebra 2 - Arithmetic Sequence Practice Name: Date: Period: A sequence is an ordered list of numbers. In an arithmetic sequence, each term is equal to the previous term, plus (or minus) a constant. The

### Double Sequences and Double Series

Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine E-mail: habil@iugaza.edu Abstract This research considers two traditional important questions,

### APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS

APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic

### Exam #5 Sequences and Series

College of the Redwoods Mathematics Department Math 30 College Algebra Exam #5 Sequences and Series David Arnold Don Hickethier Copyright c 000 Don-Hickethier@Eureka.redwoods.cc.ca.us Last Revision Date:

### P (A) = lim P (A) = N(A)/N,

1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

### Chapter 2 Sequences and Series

Chapter 2 Sequences and Series 2. Sequences A sequence is a function from the positive integers (possibly including 0) to the reals. A typical example is a n = /n defined for all integers n. The notation

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### Infinite series, improper integrals, and Taylor series

Chapter Infinite series, improper integrals, and Taylor series. Introduction This chapter has several important and challenging goals. The first of these is to understand how concepts that were discussed

### A Curious Case of a Continued Fraction

A Curious Case of a Continued Fraction Abigail Faron Advisor: Dr. David Garth May 30, 204 Introduction The primary purpose of this paper is to combine the studies of continued fractions, the Fibonacci

### 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

### Lecture 5 : The Poisson Distribution

Lecture 5 : The Poisson Distribution Jonathan Marchini November 10, 2008 1 Introduction Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,

### Taylor and Maclaurin Series

Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

### MITES Physics III Summer Introduction 1. 3 Π = Product 2. 4 Proofs by Induction 3. 5 Problems 5

MITES Physics III Summer 010 Sums Products and Proofs Contents 1 Introduction 1 Sum 1 3 Π Product 4 Proofs by Induction 3 5 Problems 5 1 Introduction These notes will introduce two topics: A notation which