Ch 9 SB answers.notebook. May 06, 2014 WARM UP

WARM UP 1

9.1 TOPICS Factorial Review Counting Principle Permutations Distinguishable permutations Combinations 2

FACTORIAL REVIEW 3

Question... How many sandwiches can you make if you have 3 types of bread, 4 types of meat and 2 types of cheese? 4

Same Question... How many sandwiches can you make if you have 3 types of bread, 4 types of meat and 2 types of cheese? 6

EXAMPLE #1 7

Permutation ways that a set of objects can be arranged 8

EXAMPLE #2 9

How many distinguishable permutations (unique arrangements) of the words below can you make? MOP MOP MPO OMP OPM POM PMO MOM MOM MMO OMM OMM MOM MMO 10

How many distinguishable permutations (unique arrangements) of the word below can you make? CANADA CANADA CANADA CANADA CANADA CANADA CANADA and more... 11

DISTINGUISHABLE PERMUTATIONS Let S be a set of n elements of k different types. Let n 1 = the number of elements of type 1, n 2 = the number of elements of type 2...n k =the number of elements of type k. Then the number of distinguishable permutations of the n elements is: 12

EXAMPLE #3 Find the number of distinguishable 13 letter "words" that can be made from the letters in MASSACHUSETTS? 13

Permutations You can also make the blanks and fill them in. It accomplishes the same result. 14

EXAMPLE #4 Your band has made 12 songs and plans to record 9 of them for a CD. In how many ways can you arrange the songs for the CD? 15

COMBINATIONS You also can still use the fill in the blank approach, you just need to make sure that you divide by the number of things that are the same. 16

EXAMPLE #5 17

Question... Can you tell the difference? Out of 20 seniors, 3 are chosen to get 3 scholarships. One is worth $1500, one is worth $1000, one is worth $500. How many ways are there to do this? Permutation or Combination? Do you care which scholarship you get??? This is called finding a permutation. The order in which each thing is picked is important. It matters if you are picked first or last. 18

Question... Can you tell the difference? does order matter? Out of 20 seniors, 3 are chosen to get 3 $500 scholarships. How many ways are there to do this? Permutation or Combination? Combination But, now there is more that you have to do. Unlike the first problem, the order in which you get picked doesn't matter. It doesn't matter if you get the 1st scholarship or the third one, you are still getting $500 This is called finding a combination. A combination is basically a permutation where it doesn't matter the order in which people were picked 19

Thinking of the same question... Suppose Tom, Mary and John win. There are 6 different ways that they can be picked: TMJ, TJM, MJT, MTJ, JMT, JTM All 6 ways have the same people winning the same scholarships, so this is the combinations of how they can be picked. So the total from before needs to be divided by 6 or (3!) That's the difference between a permutation and a combination. 20

The short and sweet version... Use Permutations when the order of selection matters Use Combinations when the order of selection does not matter 21

Practice Time 1) A company advertises two job openings, one for copy writer and one for artist. If 10 people who are qualified for either position apply, in how many ways can the opening be filled? 2) A company advertises two job openings for computer programmers, both with the same salary and job description. In how many ways can the openings be filled if 10 people apply? 22

Assignment 9.1 p.649 #1 12, 19 26, 29, 31, 33, 34 36 39, 41, 42, 45, 46, 48 (This is the assignment for Monday and Tuesday) 23

Out of 20 seniors, 3 are chosen to get 3 $500 scholarships. How many ways are there to do this? So, the solution is: SKIP This is called finding a combination. A combination is basically a permutation where it doesn't matter the order in which people were picked 24

COMBINATIONS SKIP How many permutations would there be? 25

SKIP 26

PERMUTATIONS SKIP 27

SKIP 28

QUESTION Out of 20 seniors, 3 are chosen to get 3 scholarships. One is worth $1500, one is worth $1000, one is worth $500. How many ways are there to do this? SKIP 29

Same question Out of 20 seniors, 3 are chosen to get 3 $500 scholarships. How many ways are there to do this? Start out like before doing a permutation: SKIP But, now there is more that you have to do. Unlike the first problem, the order in which you get picked doesn't matter. It doesn't matter if you get the 1st scholarship or the third one, you are still getting $500 30

SKIP 31

SKIP 32

1) Review from yesterday Suppose a club with 12 members wishes to choose a president, a vice president and a treasurer. In how many ways can this be done? 2) Suppose on the other hand that the club merely wants to choose a governing council of three. In how many ways can this be done 33

9.2 day1 TOPICS P and C backwards BINOMIAL Expansion Pascal's Triangle 34

Example #1 (REVIEW) 35

Now what...think about the set up. You WILL SEE THIS TYPE OF QUESTION AGAIN!!! Example #2 How many permutations are possible when seating 4 people around a circular table? * We need to choose where the first person sits... then from there we have 3 places to full so... 36

Example #3 CARDS How many ways are there to deal 13 cards from a standard deck of cards if the order in which the cards are dealt is (a) important? (b) not important? 37

BINOMIAL EXPANSION Expand the following: (x + y) 0 1 (x + y) 1 x + y (x + y) 2 x 2 + 2xy +y 2 38

Expand the following: (x + y) 3 (x + y)(x + y) 2 (x + y)(x 2 + 2xy + y 2 ) x 3 + 3x 2 y + 3xy 2 + y 3 39

Expand the following: (x + y) 4 (x + y) 2 (x + y) 2 (x 2 + 2xy + y 2 )(x 2 + 2xy + y 2 ) x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 40

PATTERNS...what will the next line be? (x + y) 0 1 (x + y) 1 x + y (x + y) 2 x 2 + 2xy +y 2 (x + y) 3 x 3 + 3x 2 y + 3xy 2 + y 3 (x + y) 4 x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 (x + y) 5 41

(x + y) 0 1 (x + y) 1 x + y (x + y) 2 x 2 + 2xy +y 2 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 (x + y) 3 x 3 + 3x 2 y + 3xy 2 + y 3 (x + y) 4 x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 (x + y) 5 Pascal's Triangle One use is to find the coefficients of an expansion of (x + y) n 42

nc r = = = Pascal's Triangle 43

We can use ncr to find coefficients... Example #4 Find the first 5 terms of (x + y) 20 44

Here's How... Find the first 5 terms of (x + y) 20 45

On the calculator... But what if I want all the numbers in a certain row from Pascal's Triangle so I can expand a binomial? 46

So Make a Table... 47

Assignment 9.2 p.656 #1 15 odd 48

9.2 day2 TOPICS Expansions and terms Word problem and expansion TABLE CHALLENGE 49

EXAMPLE #1 Find the fourth term of (x 2y) 10 ***Do more than one term to show alternating signs when expanded versus all "+" with (x + y) versus (x y) 50

EXAMPLE #2 Find the a 4 term of (2a b 2 ) 6 51

TABLE CHALLENGE 1) There are 14 people in a club. A committee of 6 persons are to be chosen to represent the club at a conference. In how many ways can the committee be chosen? 2) Suppose the letters in VERMONT are used to form "words". How many "words" can be formed that begin with a vowel and end with a consonant? 3) Suppose a True-False Test has 25 questions. a) In how many ways may a student mark the test if all questions are answered? b) In how many ways may a student mark the test if questions could be left blank? c) In how many ways may a student mark a test if 11 questions are answered correctly, 14 are incorrect, provided all questions are answered? 4) Expand: (x+y) 9 5) In how many ways may 9 people be seated around a circular table? 52

Remember... nc r = = 53

Assignment 9.2 p.656 #17 29 odd, 34 40 all WORKSHEET due next Thursday 54

Now for Binomial Expansion Expand:(x+y) 6 ( x + y ) 6 ( x + y ) 6 ( x + y ) 6 ( x + y ) SKIP6 55

(a + b) n = How do we use this to answer questions? Example #3 Find how many ways you can hire 20 females and 15 males. (M + F) 35 : 35 C 15 M 15 F 20 SKIP 56

SKIP 57

1) PRACTICE for QUIZ 2) 3) 4) Expand: (x + y) 9 SKIP 7 58

SKIP 59

Prove for all integers n 2 SKIP 60

Now go backwards... Example #1 Solve for n: n C 2 = 45 Solve for n: n C 2 = n 1 P 2 SKIP 61

9.3 day1 TOPICS Probability 67

Testing Your Intuition About Probability Find the probability of each of the following events. 1. Tossing a head on one toss of a fair coin 2. Tossing two heads in a row on two tosses of a fair coin 3. Drawing a queen from a standard deck of 52 cards 4. Rolling a sum of 4 on a single roll of two fair dice 5. Guessing all 6 numbers in a state lottery that requires you to pick 6 numbers between 1 and 46 inclusive 1/8,099,197,920 68

Probability of an Event (Equally Likely Outcomes) If E is an event in a finite, nonempty sample space S of equally likely outcomes, then the probablility of the event E is Probability Distribution Each outcome is assigned a unique probability 69

Probability of an Event (Equally Likely Outcomes) Find the probability of rolling a sum divisible by 3 on a single roll of two fair dice Make a sample space for sums found by rolling 2 fair dice 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 70

Probability Function (Distribution) A probability function is a function P that assigns a real number to each outcome in a sample space S subject to the following conditions: 1. 0 P(O) 1 for every outcome O 2. the sum of the probability of all outcomes in S is 1 3. P( ) = 0 EXAMPLE 1 Find the probability of rolling a sum divisible by 3 on a single roll of 2 fair dice. First find the probability distribution. Then add all your probabilities together from your favorable outcomes. Outcome 2 3 4 5 6 7 8 9 10 11 12 Probability 71

EXAMPLE 2 - Testing a Probability Function Is it possible to weight a standard 6 - sided die in such a way that the probability of rolling each number n is exactly 1/(n 2 + 1)? Hint: What would the probability distribution look like? The probability distribution would look like this: Outcomes 1 2 3 4 5 6 Probabilities 1/2 1/5 1/10 1/17 1/26 1/37 This is not a valid probability function because the probabilities do not equal 1 72

Probability of an Event (Outcomes Not Equally Likely) Let S be a finite, nonempty space in which every outcome has a probability assigned to it by a probability function P. If E is any event in S, the probability of the event E is the sum of the probabilities of all outcomes contained in E. 73

Strategy for Determining Probabilities 1. Determine the sample space of all possible outcomes. When possible choose outcomes that are equally likely 2. If the sample space has equally likely outcomes, the probability of an event E is determined by counting: 3. If the sample space does not have equally likely outcomes, determine the probability function. Check to be sure that the conditions of a probability function are satisfied. Then the probability of an event E is determined by adding up the probabilities of all the outcomes contained in E 74

Multiple Principle of Probability Suppose an event A has probability p 1 and an event B has probability p 2 under the assumption that A occurs. Then the probability that both A and B occur is p 1 p 2 EXAMPLE 3 Sal opens a box of a dozen cremes and generously offers two of them to Val. Val likes vanilla cremes the best, but all the chocolates look alike on the outside. If four of the twelve cremes are vanilla, what is the probability that both of Val's picks turn out to be vanilla? 75

Probability using Combinations (Instead of Multiplication Principle) EXAMPLE 3: Same question, different method: Sal opens a box of a dozen cremes and generously offers two of them to Val. Val likes vanilla cremes the best, but all the chocolates look alike on the outside. If four of the twelve cremes are vanilla, what is the probability that both of Val's picks turn out to be vanilla? Remember: From the four vanilla cremes, Val wants 2 so 4 C 2 = 6 From the box of 12, Val chooses 2 so 12 C 2 = 66 Therefore P(E) = 6/66 or 1/11 76

Venn Diagrams EXAMPLE 4: In a large high school, 54% of the students are girls and 62% of the students play sports. Half of the girls play sports. a) What percentage of the students who play sports are boys? b) If a student is chosen at random, what is the probability that it is a boy who does not play sports? 77

EXAMPLE 5 Tree Diagrams Two identical cookie jars are on a counter. Jar A contains 2 chocolate chip cookies and 2 peanut butter cookies, while Jar B contains 1 chocolate chip cookie. We select a cookie at random. What is the probability that it is a chocolate chip cookie? 0.5 Jar A 0.25 0.25 0.25 0.25 CC CC PB PB P(choc chip) = 0.5 Jar B 1 CC 78

Conditional Probability Formula If the event B depends on the event A, then EXAMPLE 6 Two identical cookie jars are on a counter. Jar X contains 2 chocolate chip and 2 peanut butter cookies, while jar Y contains 1 chocolate chip cookie. Suppose you draw a cookie at random from one of the jars. Given that it is a chocolate chip cookie, what is the probability it came from jar X? 79

EXAMPLE 6 Solution Two identical cookie jars are on a counter. Jar X contains 2 chocolate chip and 2 peanut butter cookies, while jar Y contains 1 chocolate chip cookie. Suppose you draw a cookie at random from one of the jars. Given that it is a chocolate chip cookie, what is the probability it came from jar X? 0.5 Jar X 0.25 0.25 0.25 0.25 CC CC PB PB P( chocolate chip) 0.5 Jar Y 1 CC 80

Assignment 9.3 p.666 #1 27 odd WORKSHEET Permutation and Combinations due Thursday 81

9.3 day2 TOPICS Probability and Binomial Expansions 82

Remember Binomial Expansion from last week? (a + b) n = How do we use this to answer questions? With questions such as: Find how many ways you can hire 20 females and 15 males. (M + F) 35 : 35 C 15 M 15 F 20 83

Binomial Distributions (REALLY IMPORTANT NOTES) For events that have 2 possible outcomes (example: heads or tails) we can use the Binomial theorem that we learned last section to determine probabilities. (X + Y) n number of trials Probability of each outcome, ALWAYS sum to 1. 84

EXAMPLE 1 Suppose you roll a die 4 times. Find the probability that you roll exactly two 3's. Let x = probability of rolling a 3 (which is what??) Let y = probability of not rolling a 3 (which is what??) The probability that you need can be found by looking at the binomial expansion of (x + y) 4 Remember that (x + y) 4 = 4 C 0 x 4 + 4 C 1 x 3 y + 4 C 2 x 2 y 2 + 4 C 3 xy 3 + 4 C 4 y 4 = x 4 + 4x 3 y + 6x 2 y 2 +4xy 3 + y 4 We are looking for the term x 2 y 2 (2 3's and 2 not 3's) 4C 2 x 2 y 2 P(exactly 2 3's) = 6(1/6) 2 (5/6) 2 = 0.11574 85

EXAMPLE 2 Suppose you roll a die 4 times. What is the probability of rolling all 3's? EXAMPLE 3 Suppose you roll a die 4 times. What is the probability of rolling at least 2 3's? 86

EXAMPLE 4 (A,B, and C) Suppose Michael makes 90% of his free throws. If he shoots 20 free throws, and if his chances of making each one is independent of the other shots, what is the probability he makes: A) All 20 B) Exactly 18 C) At least 18 87

Assignment 9.3 p.666 #28, 31, 33, 35, 39, 41 53 56 Add today's examples into your Stenos AND expand (2x 3y) 4 WORKSHEET Permutation and Combinations due Thursday 89

Probability outcome: a possible result of an experiment event: is an outcome sample space: set of ALL possible outcomes What is the sample space of flipping a coin and rolling a die? Probability of an event As a % : 0-100% As a # : 0-1 P(event) The measure of the likelihood, or chance, that the event WILL occur. Can be expressed as a decimal, fraction, or percent. 25% - unlikely 50% - equally likely to happen or not happen 75% - likely 100% - certain 90

2 Types of probability what SHOULD happen Theoretical Probability = # of favorable outcomes Total # of outcomes what happened after actually doing the experiment Experimental Probability = # of successes # of trials 91

What is the Theoretical Probability that the spinner lands on blue? P(blue) = What is the Experimental Probability that the spinner lands on blue? P(blue) = 92

You need Combination formula 94

9.4 TOPICS Arithmetic Sequences Geometric Sequences Explicit Recursive Diverge vs Converge 97

Notations you need to understand: a 1 a n a n 1 n Recursive Formula: Formula based on knowing the previous term. NOW = NEXT EXPLICIT Formula: Formula used to find the VALUE of a term knowing which term you are testing. Find the n th term 98

explicit starting term How do you find the common difference? value of that term n th term difference between the values 99

Practice 1. Find the first 3 terms in the sequence where a 1 =6 and d=2.5 2. Find the nth term of the following sequence: a 1 = 5, d= 1/2, n=13 3. 203 is the th term of 5, 1, 3... 4. Find the arithmetic means in the sequence: 21,,, 18, 100

explicit formula is How do you calculate the ratio? 101

Practice 1. Find the first 5 terms in the sequence where a 1 =4 and r=1/2 2. Find the nth term of the following sequence: a 1 = 5, r= 3, n=7 3. 62,500 is the th term of 4, 20, 100... 102

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Arithmetic, Geometric or Neither 3, 8, 13, 18... 4, 8, 16, 32... 1, 3, 5, 7,... 27, 18, 12, 8,... 2, 5, 7, 10... 104

Explicit t n is given in terms of n. Do not need to know any of the previous terms to find a specific term Arithmetic Explicit Explicit or Recursive Geometric Explicit 105

Recursive a formula that gives a rule for t n in terms of t n 1 Recursive definition consists of two parts: An initial condition that tells where the sequence starts An equation that tells how any term in the sequence is related to the preceding term For each of the recursive formulas given below, given the first 4 terms of the sequence and write an explicit formula for the sequence if possible t 1 = 3 t n = 2t n 1 + 1 t 1 = 23 t n = t n 1 3 t 1 = 64 t n =.5t n 1 106

EXAMPLE #1 The fourth and seventh terms of an arithmetic sequence are 8 and 4, respectively. Find the first term and a recursive rule for the nth term. 107

EXAMPLE #2 The second and eighth terms of a geometric sequence are 3 and 192, respectively. Find the first term, the common ratio and the explicit rule for the nth term. 108

EXAMPLE #3 The population of a city is growing at a rate of 15% each year. How long before the city triples in size? 109

THOUGHT Is the sequence 2, 2, 2, 2,... Geometric, Arithmetic, Neither or Both? 110

Assignment 9.4 p.676 #1 20, 21 33 mod 3 37, 39, 43 48 WORKSHEET due TOMORROW 111

Practice Questions for Quiz 1. If it rains tomorrow, the probability that John will practice piano is 0.8. If it doesn't rain, there is only a 0.4 chance that John will practice. Suppose the chance of rain is 60% tomorrow. What is the probability that john will practice? 2. Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how many students are in neither class? How many are in either class? 3. You toss a coin 15 times. What is the probability you get 9 heads and 6 tails? 4. A jar has 5 blue marbles, 7 red marbles and 9 yellow marbles. What is the probability of drawing a blue and then a yellow if (a) the first marble is not replaced before drawing the second, (b) the first is put back before drawing the second? 5. Find the first 8 terms of the recursively defined sequence: t 1 = 5, t n = 3(t n-1 ) - 2 6. Find the first 4 terms of the explicitly defined sequence: t n = n 2 + 1 112

Practice Questions for Quiz 1. If the school cafeteria serves meat loaf, there is a 70% chance that they will serve peas. If they do not serve meat loaf, there is a 30% chance they will serve peas anyway. Meat loaf will be served exactly once during the 5- day week. If you are going to eat lunch today, what is the probability that peas are on the menu? 2. A survey of 1000 people was conducted by a local ice cream shop. Two questions were asked: Do you like chocolate? Do you like vanilla? The probability of a person liking chocolate is 45%. The probability of a person liking vanilla is 52%. 50% of the people who like vanilla also like chocolate. How many people do not like either chocolate or vanilla? 3. You toss a coin 18 times. What is the probability you get 12 heads and 6 tails? 4. If the track is wet, Champion has a 70% chance of winning the fifth race at Belmont. If the track is dry, she only has a 40% chance of winning. Weather forecasts an 80% chance the track will be wet. Find the probability that Chamion wins the race. 5. Find the first 4 terms of the recursively defined sequence: t 1 = -1, t n = (t n-1 ) 2 + 2 6. Find the first 4 terms of the explicitly defined sequence: t n = 4n + 3 114

EXAMPLES 116

EXAMPLE The second and eighth terms of a geometric sequence are 3 and 192, respectively. Find the first term, the common ratio and the explicit rule for the nth term. 117

EXAMPLE The fourth and seventh terms of an arithmetic sequence are 8 and 4, respectively. Find the first term and a recursive rule for the nth term. 118

EXAMPLE The population of a certain country grows as a result of two conditions: 1) The annual population growth is 1% of those already in the country 2) 20,000 people immigrate into the country each year If the population is now 5,000,000 people, what will the population be in two years? In 20 years? 119

9.5 day1 TOPICS Review of 9.4 Finite Series Sums of Finite Series 120

EXAMPLE #1 (review) The third and 8th terms of an arithmetic sequence are 1 and 11, respectively. Find the first term, the common difference, and a explicit rule for the nth term. 121

EXAMPLE #2 3 (REVIEW) #2. #3. 122

#2. EXAMPLE #2 3 (REVIEW) #3. 123

Summation Notation 124

Example #4 Find the SUM of each: A) B) C) D) 125

Example #5 A) Express each of the following series using sigma notation 3 + 6 + 9 + 12 +...+ 300 B) 1 + 8 + 27 + 64 + 125 C) D) 126

If you know: n = # of terms a 1 = 1st term a n = Last term If you know: n = # of terms a 1 = 1st term d = common difference 127

EXAMPLES #6 and #7- Find the sum of the arithmetic sequences. 6. -7, -3, 1, 5, 9, 13 7. 117, 110, 103,..., 33 128

If you know: n = # of terms a 1 = 1st term r = ratio EXAMPLES #8 and #9 - Find the sum of the geometric sequence. 8. 5, 15, 45,..., 98,145 9. 42, -7, 7/6,..., 42(-1/6) 9 129

Assignment 9.5 p.684 #3 23, mod 3, 35, 36, 37 From 9.5 Day 1 Choose 1 from 1 3, 2 from Ex 4, 2 from Ex 5, 6, 8, 9 130

Do you have any questions? 131

9.5 day2 TOPICS Converge and Diverge (9.4) Infinite Series Sums of Infinite Series 132

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What would the sum of the series be if the series had infinite terms? 2 + 4 + 8 + 16 + 32....1 +.01 +.001 +.0001... The first series diverges because as you add more terms the sum gets bigger with no end The second series converges because eventually you will add such a small amount that the sum does not get bigger How do you tell if a series converges or diverges? 134

Geometric Series IMPORTANT 135

EXAMPLE #1 136

EXAMPLES Does the following series diverge or converge? If it converges, give its sum 2) 6 + 3 + 3/2 + 3/4 +... 3) 1/48 + 1/16 + 3/16 + 9/16 +... 4) 4-2 + 1-1/2 + 1/4... 137

EXAMPLE #5-- WORD PROBLEM A ball is dropped from a height of 5 feet. Each time it rebounds to 80% of the last bounce. What is the total distance that the ball travels before coming to rest? 138

REPEATING DECIMALS Express 0. 12121212... as a fraction First, write as a series:.12 +.0012 +.000012 +... Then determine r r =.01 Then find the "sum" a 1 - r 139

REPEATING DECIMALS...again Express 0. 12121212... as a fraction n =.1212121212... 100n = 12.12121212... - n = 0.12121212... 99n = 12 n = 12/99 140

EXAMPLE #6 Write.234234234... as a fraction 141

Assignment 9.5 p.684 #25 32, 39, 41 46 Stenoes: Today: Pick 2 from #1 4 and 5 & 6 From 9.5 Day 1(Yesterday) Choose 1 from 1 3, 2 from Ex 4, 2 from Ex 5, 6, 8, 9 142

State the first 4 terms of the sequence, then state if the sequence is arithmetic, geometric or neither. t n = 5n +2 t n = 3 n t n = n 3 t n = REVIEW and WARM UP 143

Give each series in expanded form k 144

EXAMPLES - Find the sum of the first n terms. The sequence is either arithmetic or geometric. 1. 2, 5, 8,... ; n = 10 2. 4, -2, 1, -.5,..., ; n = 12 3. 1, 2, 3,..., ; n = 500 145

Today's Plan Probability Worksheet - Due TODAY!!! Series and Sequence Worksheet - Due Friday Notebook Assignments - Due Wednesday Study for TEST!!! - material on web page 146

Today's Plan Series and Sequence Worksheet - Due Friday Standard Prep 9 -- Due Wednesday Notebook Assignments - Due Wednesday Study for TEST!!! - material on web page 147

Today's Plan Series and Sequence Worksheet - Due TODAY TAKE HOME QUIZ - DUE MONDAY!! Standard Prep 9 -- Due Wednesday Notebook Assignments - Due Wednesday Study for TEST!!! - material on web page 148

Today's Plan TAKE HOME QUIZ - DUE TODAY!! Standard Prep 9 -- Due Wednesday Notebook Assignments - Due Wednesday Study for TEST!!! - material on web page Projects Assigned tomorrow - find a group! 149

Today's Plan Standard Prep 9 -- Due Wednesday Notebook Assignments - Due Wednesday Study for TEST!!! - material on web page Projects Assigned TODAY!!! Know your group! 150

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Converge vs Diverge Let {a n } be a sequence of real numbers and consider If the limit is a finite number L, the sequences converges and L is the limit of the sequences. If the limit is infinite or nonexistent, the sequence diverges Examples: 153