CHAPTER 2 SEQUENCES AND SERIES OUTLINE Geometric Series Infinite Geometric Series and Sigma Notation 4 Review

11.1 Arithmetic Sequences and Series

11.1 Arithmetic Sequeces ad Series A itroductio 1, 4, 7, 10, 13 9, 1, 7, 15 6., 6.6, 7, 7.4 ππ+, 3, π+ 6 Arithmetic Sequeces ADD To get ext term 35 1 7. 3π + 9, 4, 8, 16, 3 9, 3, 1, 1/ 3 1,1/ 4,1/16,1/

is also known as the general term of the sequence

Lesso : Sequeces ad Series Outlie Objectives: I ca determie whether a sequece has a patter. I ca determie whether a sequece ca be geeralized to fid a formula for the geeral term i the sequece. I ca determie

Section 6.4: Series. Section 6.4 Series 413

ectio 64 eries 4 ectio 64: eries A couple decides to start a college fud for their daughter They pla to ivest $50 i the fud each moth The fud pays 6% aual iterest, compouded mothly How much moey will they

T1.1 Lesson 3 - Arithmetic & Geometric Series & Summation Notation

Fast Five T. Lesso 3 - Arithmetic & Geometric eries & ummatio Notatio Math L - atowski Fid the sum of the first 00 umbers Outlie a way to solve this problem ad the carry out your pla Fid the sum of the

Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or

Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad

Ma 530 Infinite Series I

Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li

SEQUENCES AND SERIES

Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces

10.2 Infinite Series Contemporary Calculus 1

10. Ifiite Series Cotemporary Calculus 1 10. INFINITE SERIES Our goal i this sectio is to add together the umbers i a sequece. Sice it would take a very log time to add together the ifiite umber of umbers,

Unit 6: Sequences and Series

AMHS Hoors Algebra 2 - Uit 6 Uit 6: Sequeces ad Series 26 Sequeces Defiitio: A sequece is a ordered list of umbers ad is formally defied as a fuctio whose domai is the set of positive itegers. It is commo

Further Exploration of Patterns

Further Exploratio of Patters Abstract Quadratic Patters a+b+c 4a+b+c 9a+3b+c 16a+4b+c 5a+5b+c 1 st chage 3a+b 5a+b 7a+b 9a+b d chage a a a 1 Coefficiet of is ( a) a. Costat secod chage, therefore it is

Section 7 Fundamentals of Sequences and Series

ectio Fudametals of equeces ad eries. Defiitio ad examples of sequeces A sequece ca be thought of as a ifiite list of umbers. 0, -, -0, -, -0...,,,,,,. (iii),,,,... Defiitio: A sequece is a fuctio which

MISCELLANEOUS SEQUENCES & SERIES QUESTIONS

MISCELLANEOUS SEQUENCES & SERIES QUESTIONS Questio (***+) Evaluate the followig sum 30 r ( 2) 4r 78. 3 MP2-V, 75,822,200 Questio 2 (***+) Three umbers, A, B, C i that order, are i geometric progressio

Infinite Sequences and Series

Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

CHAPTER 10 INFINITE SEQUENCES AND SERIES

CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece

Infinite Series. Definition. An infinite series is an expression of the form. Where the numbers u k are called the terms of the series.

Ifiite Series Defiitio. A ifiite series is a expressio of the form uk = u + u + u + + u + () 2 3 k Where the umbers u k are called the terms of the series. Such a expressio is meat to be the result of

Sequences. A Sequence is a list of numbers written in order.

Sequeces A Sequece is a list of umbers writte i order. {a, a 2, a 3,... } The sequece may be ifiite. The th term of the sequece is the th umber o the list. O the list above a = st term, a 2 = 2 d term,

VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)

VERY SHORT ANSWER TYPE QUESTIONS ( MARK). If th term of a A.P. is 6 7 the write its 50 th term.. If S = +, the write a. Which term of the sequece,, 0, 7,... is 6? 4. If i a A.P. 7 th term is 9 ad 9 th

Math 113 Exam 3 Practice

Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you

Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018

CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical

MA131 - Analysis 1. Workbook 7 Series I

MA3 - Aalysis Workbook 7 Series I Autum 008 Cotets 4 Series 4. Defiitios............................... 4. Geometric Series........................... 4 4.3 The Harmoic Series.........................

Series Review. a i converges if lim. i=1. a i. lim S n = lim i=1. 2 k(k + 2) converges. k=1. k=1

Defiitio: We say that the series S = Series Review i= a i is the sum of the first terms. i= a i coverges if lim S exists ad is fiite, where The above is the defiitio of covergece for series. order to see

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

Essential Question How can you recognize an arithmetic sequence from its graph?

. Aalyzig Arithmetic Sequeces ad Series COMMON CORE Learig Stadards HSF-IF.A.3 HSF-BF.A. HSF-LE.A. Essetial Questio How ca you recogize a arithmetic sequece from its graph? I a arithmetic sequece, the

Sequences. Notation. Convergence of a Sequence

Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it

1 Generating functions for balls in boxes

Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways

Polynomial Functions and Their Graphs

Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively

a n b. n a n Test Review for Lessons 9.1 and 9.3 Show all work on a separate sheet of paper for full credit.

Precalculus Test Review for Lessos. ad. Name: Date: Hour: Show all work o a separate sheet of paper for full credit.. Write the first four terms of the sequeces give by a. a b. a 7. Write the first five

MAT 271 Project: Partial Fractions for certain rational functions

MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,

6.3 Testing Series With Positive Terms

6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1

September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright

Series: Infinite Sums

Series: Ifiite Sums Series are a way to mae sese of certai types of ifiitely log sums. We will eed to be able to do this if we are to attai our goal of approximatig trascedetal fuctios by usig ifiite degree

The Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1

460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that

RADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify

Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL

Roberto s Notes on Infinite Series Chapter 1: Sequences and series Section 3. Geometric series

Roberto s Notes o Ifiite Series Chapter 1: Sequeces ad series Sectio Geometric series What you eed to kow already: What a ifiite series is. The divergece test. What you ca le here: Everythig there is to

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

Define and Use Sequences and Series

. a., A..A; P..A, P..B TEKS Defie ad Use Sequeces ad Series Before You idetified ad wrote fuctios. Now You will recogize ad write rules for umber patters. Why? So you ca fid agle measures, as i Ex.. Key

Math 2412 Review 3(answers) kt

Math 4 Review 3(aswers) kt A t A e. If the half-life of radium is 690 years, ad you have 0 grams ow, how much will be preset i 50 years (rouded to three decimal places)?. The decay of radium is modeled

MIXED REVIEW of Problem Solving

MIXED REVIEW of Problem Solvig STATE TEST PRACTICE classzoe.com Lessos 2.4 2.. MULTI-STEP PROBLEM A ball is dropped from a height of 2 feet. Each time the ball hits the groud, it bouces to 70% of its previous

Math 113 Exam 4 Practice

Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for

Sequences, Sums, and Products

CSCE 222 Discrete Structures for Computig Sequeces, Sums, ad Products Dr. Philip C. Ritchey Sequeces A sequece is a fuctio from a subset of the itegers to a set S. A discrete structure used to represet

x a x a Lecture 2 Series (See Chapter 1 in Boas)

Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio

SEQUENCES AND SERIES

9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first

Chapter 6: Numerical Series

Chapter 6: Numerical Series 327 Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals

Progressions. ILLUSTRATION 1 11, 7, 3, -1, i s an A.P. whose first term is 11 and the common difference 7-11=-4.

Progressios SEQUENCE A sequece is a fuctio whose domai is the set N of atural umbers. REAL SEQUENCE A Sequece whose rage is a subset of R is called a real sequece. I other words, a real sequece is a fuctio

Infinite Sequence and Series

Chapter 7 Ifiite Sequece ad Series 7. Sequeces A sequece ca be thought of as a list of umbers writte i a defiite order: a,a 2,a 3,a 4,...,a,... The umber a is called the first term, a 2 is the secod term,

= = =

Sec 5.8 Sec 6. Mathematical Modelig (Arithmetic & Geometric Series) Name: Carl Friedrich Gauss is probably oe of the most oted complete mathematicias i history. As the story goes, he was potetially recogiized

Alternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n.

0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Sectio 9.5 Alteratig Series Use the Alteratig Series Test to determie whether a ifiite series coverges. Use the Alteratig Series Remaider to approximate

Infinite Sequences and Series

-3 OJECTIVES Fid the it of the terms of a ifiite sequece. Fid the sum of a ifiite geometric series. Ifiite Sequeces ad Series ECONOMICS O Jauary 8, 999, Florida goveror Jeb ush proposed a tax cut that

Chapter 10: Power Series

Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because

P.3 Polynomials and Special products

Precalc Fall 2016 Sectios P.3, 1.2, 1.3, P.4, 1.4, P.2 (radicals/ratioal expoets), 1.5, 1.6, 1.7, 1.8, 1.1, 2.1, 2.2 I Polyomial defiitio (p. 28) a x + a x +... + a x + a x 1 1 0 1 1 0 a x + a x +... +

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =!

.8,.9: Taylor ad Maclauri Series.8. Although we were able to fid power series represetatios for a limited group of fuctios i the previous sectio, it is ot immediately obvious whether ay give fuctio has

If we want to add up the area of four rectangles, we could find the area of each rectangle and then write this sum symbolically as:

Sigma Notatio: If we wat to add up the area of four rectagles, we could fid the area of each rectagle ad the write this sum symbolically as: Sum A A A A Liewise, the sum of the areas of te triagles could

Chapter 6 Overview: Sequences and Numerical Series. For the purposes of AP, this topic is broken into four basic subtopics:

Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals (which is what most studets

Chapter 7: Numerical Series

Chapter 7: Numerical Series Chapter 7 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals

Ma 530 Introduction to Power Series

Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power

Chapter 9 Sequences, Series, and Probability Section 9.4 Mathematical Induction

Chapter 9 equeces, eries, ad Probability ectio 9. Mathematical Iductio ectio Objectives: tudets will lear how to use mathematical iductio to prove statemets ivolvig a positive iteger, recogize patters

, 4 is the second term U 2

Balliteer Istitute 995-00 wwwleavigcertsolutioscom Leavig Cert Higher Maths Sequeces ad Series A sequece is a array of elemets seperated by commas E,,7,0,, The elemets are called the terms of the sequece

AP Calculus Chapter 9: Infinite Series

AP Calculus Chapter 9: Ifiite Series 9. Sequeces a, a 2, a 3, a 4, a 5,... Sequece: A fuctio whose domai is the set of positive itegers = 2 3 4 a = a a 2 a 3 a 4 terms of the sequece Begi with the patter

a 3, a 4, ... are the terms of the sequence. The number a n is the nth term of the sequence, and the entire sequence is denoted by a n

60_090.qxd //0 : PM Page 59 59 CHAPTER 9 Ifiite Series Sectio 9. EXPLORATION Fidig Patters Describe a patter for each of the followig sequeces. The use your descriptio to write a formula for the th term

SEQUENCE AND SERIES NCERT

9. Overview By a sequece, we mea a arragemet of umbers i a defiite order accordig to some rule. We deote the terms of a sequece by a, a,..., etc., the subscript deotes the positio of the term. I view of

10.6 ALTERNATING SERIES

0.6 Alteratig Series Cotemporary Calculus 0.6 ALTERNATING SERIES I the last two sectios we cosidered tests for the covergece of series whose terms were all positive. I this sectio we examie series whose

Math 113 Exam 3 Practice

Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how

NUMERICAL METHODS COURSEWORK INFORMAL NOTES ON NUMERICAL INTEGRATION COURSEWORK

NUMERICAL METHODS COURSEWORK INFORMAL NOTES ON NUMERICAL INTEGRATION COURSEWORK For this piece of coursework studets must use the methods for umerical itegratio they meet i the Numerical Methods module

MATH 2300 review problems for Exam 2

MATH 2300 review problems for Exam 2. A metal plate of costat desity ρ (i gm/cm 2 ) has a shape bouded by the curve y = x, the x-axis, ad the lie x =. (a) Fid the mass of the plate. Iclude uits. Mass =

The z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j

The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.

Permutations, Combinations, and the Binomial Theorem

Permutatios, ombiatios, ad the Biomial Theorem Sectio Permutatios outig methods are used to determie the umber of members of a specific set as well as outcomes of a evet. There are may differet ways to

Once we have a sequence of numbers, the next thing to do is to sum them up. Given a sequence (a n ) n=1

. Ifiite Series Oce we have a sequece of umbers, the ext thig to do is to sum them up. Give a sequece a be a sequece: ca we give a sesible meaig to the followig expressio? a = a a a a While summig ifiitely

= n which will be written with general term a n Represent this sequence by listing its function values in order:

Sectio 9.: Sequeces a Series (Sums) Remier:, 2, 3,, k 2, k, k, k +, k + 2, The ellipsis ots iicate the sequece is oeig, has ifiitely may terms. I. SEQUENCES Defiitio (page 706): A sequece is a fuctio whose

ARRANGEMENTS IN A CIRCLE

ARRANGEMENTS IN A CIRCLE Whe objects are arraged i a circle, the total umber of arragemets is reduced. The arragemet of (say) four people i a lie is easy ad o problem (if they liste of course!!). With

INFINITE SEQUENCES AND SERIES

11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS

UNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series

UNIT #5 SEQUENCES AND SERIES Lesso # Sequeces Lesso # Arithmetic ad Geometric Sequeces Lesso #3 Summatio Notatio Lesso #4 Arithmetic Series Lesso #5 Geometric Series Lesso #6 Mortgage Paymets COMMON CORE

Chapter 6 Infinite Series

Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat

Quiz No. 1. ln n n. 1. Define: an infinite sequence A function whose domain is N 2. Define: a convergent sequence A sequence that has a limit

Quiz No.. Defie: a ifiite sequece A fuctio whose domai is N 2. Defie: a coverget sequece A sequece that has a limit 3. Is this sequece coverget? Why or why ot? l Yes, it is coverget sice L=0 by LHR. INFINITE

Math 155 (Lecture 3)

Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,

Lecture 2.5: Sequences

Lecture.5: Sequeces CS 50, Discrete Structures, Fall 015 Nitesh Saxea Adopted from previous lectures by Zeph Gruschlag Course Admi HW posted Covers Chapter Due Oct 0 (Tue) Mid Term 1: Oct 15 (Thursday)

In number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.

Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special

XT - MATHS Grade 12. Date: 2010/06/29. Subject: Series and Sequences 1: Arithmetic Total Marks: 84 = 2 = 2 1. FALSE 10.

ubject: eries ad equeces 1: Arithmetic otal Mars: 8 X - MAH Grade 1 Date: 010/0/ 1. FALE 10 Explaatio: his series is arithmetic as d 1 ad d 15 1 he sum of a arithmetic series is give by [ a ( ] a represets

7 Sequences of real numbers

40 7 Sequeces of real umbers 7. Defiitios ad examples Defiitio 7... A sequece of real umbers is a real fuctio whose domai is the set N of atural umbers. Let s : N R be a sequece. The the values of s are

The Phi Power Series

The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at http://semresearch.wordpress.com/.

SEQUENCES AND SERIES

Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9. Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say that

Starting with a time of 1 second, the partial sums of the time series form the sequence 1, 3 2, 7 4, 1 5

- OBJECTIVE Determie whether a series is coverget or diverget. Coverget ad Diverget Series HISTORY The Greek philosopher Zeo of Elea (c. 90 30 B.C.) proposed several perplexig riddles, or paradoxes. Oe

MIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS

MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will

Eton Education Centre JC 1 (2010) Consolidation quiz on Normal distribution By Wee WS (wenshih.wordpress.com) [ For SAJC group of students ]

JC (00) Cosolidatio quiz o Normal distributio By Wee WS (weshih.wordpress.com) [ For SAJC group of studets ] Sped miutes o this questio. Q [ TJC 0/JC ] Mr Fruiti is the ower of a fruit stall sellig a variety

Exponents. Learning Objectives. Pre-Activity

Sectio. Pre-Activity Preparatio Epoets A Chai Letter Chai letters are geerated every day. If you sed a chai letter to three frieds ad they each sed it o to three frieds, who each sed it o to three frieds,

Part I: Covers Sequence through Series Comparison Tests

Part I: Covers Sequece through Series Compariso Tests. Give a example of each of the followig: (a) A geometric sequece: (b) A alteratig sequece: (c) A sequece that is bouded, but ot coverget: (d) A sequece

A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence

Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as

Power and Type II Error

Statistical Methods I (EXST 7005) Page 57 Power ad Type II Error Sice we do't actually kow the value of the true mea (or we would't be hypothesizig somethig else), we caot kow i practice the type II error

Quadratic Functions. Before we start looking at polynomials, we should know some common terminology.

Quadratic Fuctios I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively i mathematical

= 4 and 4 is the principal cube root of 64.

Chapter Real Numbers ad Radicals Day 1: Roots ad Radicals A2TH SWBAT evaluate radicals of ay idex Do Now: Simplify the followig: a) 2 2 b) (-2) 2 c) -2 2 d) 8 2 e) (-8) 2 f) 5 g)(-5) Based o parts a ad

Power Series Expansions of Binomials

Power Series Expasios of Biomials S F Ellermeyer April 0, 008 We are familiar with expadig biomials such as the followig: ( + x) = + x + x ( + x) = + x + x + x ( + x) 4 = + 4x + 6x + 4x + x 4 ( + x) 5

FLC Ch 8 & 9. Evaluate. Check work. a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) 3. p) q) r) s) t) 3.

Math 100 Elemetary Algebra Sec 8.1: Radical Expressios List perfect squares ad evaluate their square root. Kow these perfect squares for test. Def The positive (pricipal) square root of x, writte x, is

4x 2. (n+1) x 3 n+1. = lim. 4x 2 n+1 n3 n. n 4x 2 = lim = 3

Exam Problems (x. Give the series (, fid the values of x for which this power series coverges. Also =0 state clearly what the radius of covergece is. We start by settig up the Ratio Test: x ( x x ( x x

Q.11 If S be the sum, P the product & R the sum of the reciprocals of a GP, find the value of

Brai Teasures Progressio ad Series By Abhijit kumar Jha EXERCISE I Q If the 0th term of a HP is & st term of the same HP is 0, the fid the 0 th term Q ( ) Show that l (4 36 08 up to terms) = l + l 3 Q3

Solving equations (incl. radical equations) involving these skills, but ultimately solvable by factoring/quadratic formula (no complex roots)

Evet A: Fuctios ad Algebraic Maipulatio Factorig Square of a sum: ( a + b) = a + ab + b Square of a differece: ( a b) = a ab + b Differece of squares: a b = ( a b )(a + b ) Differece of cubes: a 3 b 3

The Comparison Tests. Examples. math 131 infinite series, part iii: comparison tests 18

math 3 ifiite series, part iii: compariso tests 8 The Compariso Tests The idea behid the compariso tests is pretty simple. Suppose we have a series such as which we kow coverges by the p-series test. Now

MA131 - Analysis 1. Workbook 3 Sequences II

MA3 - Aalysis Workbook 3 Sequeces II Autum 2004 Cotets 2.8 Coverget Sequeces........................ 2.9 Algebra of Limits......................... 2 2.0 Further Useful Results........................

CHAPTER - 9 SEQUENCES AND SERIES KEY POINTS A sequece is a fuctio whose domai is the set N of atural umbers. A sequece whose rage is a subset of R is called a real sequece. Geeral A.P. is, a, a + d, a

Section 5.5. Infinite Series: The Ratio Test

Differece Equatios to Differetial Equatios Sectio 5.5 Ifiite Series: The Ratio Test I the last sectio we saw that we could demostrate the covergece of a series a, where a 0 for all, by showig that a approaches