BC Sequences and series 2015.notebook March 03, 2015

Transcription

1 Feb 25 12:49 PM 1

2 Jan 27 9:13 AM 2

3 Jan 27 9:24 AM 3

4 Feb 25 12:49 PM 4

5 Warm up thoughts What notation is used for sequences? What notation is used for series? How are they related? Jan 13 8:01 AM 5

6 Limit Facts this means that when the input n goes into the function f, it generates the elements of the series in other words, if you have a function that will generate the elements of the series, then the limit is the same as the function Feb 24 3:57 PM 6

7 Warm up Random free response question from AB 15 minutes to solve (alone) Feb 23 12:20 PM 7

8 Warm up < 15 minutes Feb 23 8:30 AM 8

9 Feb 23 8:30 AM 9

10 Feb 23 12:20 PM 10

11 Feb 23 12:19 PM 11

12 Feb 23 12:06 PM 12

13 Feb 23 1:21 PM 13

14 definitions you will need to understand sequence - a list of ordered numbers separated by commas series - adds the terms of a sequence together partial sums - SOP--adds the a few terms of a sequence monotonic - every term is increasing or every term is decreasing --all one behavior bounded - trapped by a certain value -- "ceiling" on function converge - has a numerical "limit"...headed towards some value diverge - either oscillates or heads off to infinity Feb 24 9:29 PM 14

15 Limit Facts this will lead us to the idea of absolute convergence and conditional convergence In other words: if the limit goes to zero even if you ignore any negative terms the series is "absolutely" convergent if it converges only if some of the terms are negative, then it is "conditionally" convergent Feb 24 4:08 PM 15

16 1st Step: Nth Term Divergence Test limit of the sequence the series but... it will take a little more work to prove Feb 24 3:09 PM 16

17 Jan 13 9:08 AM 17

18 Feb 23 12:42 PM 18

19 Review activity when bell rings Feb 24 12:42 PM 19

20 1st Step: Nth Term Divergence Test limit of the sequence the series but... it will take a little more work to prove Feb 24 12:41 PM 20

21 Feb 23 12:42 PM 21

22 Once we decide it MIGHT converge we need to look for a few "recognizable" series. They are : 1. Geometric (we can actually find the sum of this one) 2. Telescoping (we can actually find the sum of this one) 3. P-series special case harmonic 4. Alternating You need to memorize these series Feb 24 3:44 PM 22

23 Geometric Series Can be written in the form notice this in one of the only ones that starts at zero instead of 1 Conditions: if if the series converges to the series diverges Feb 24 3:42 PM 23

24 ex. 1 Feb 24 4:20 PM 24

25 ex. 2 Feb 23 12:13 PM 25

26 ex. 3 Jan 13 9:19 AM 26

27 Telescoping Series Can be written in the form Characteristics: *some of the terms begin to cancel each other out leaving only a finite number of terms remaining * you will need to calculate a few of the terms to decide which ones will remain Feb 24 3:42 PM 27

28 ex 1 nth div test first n=1 n=2 n=3 n=4 n=5 Feb 24 6:58 PM 28

29 telescoping often shows up looking like this ex. 2 can be rewritten using partial fractions Feb 24 7:42 PM 29

30 Jan 14 8:50 AM 30

31 Feb 24 7:48 PM 31

32 Warm ups from the note packet first Hint: nth divergence test second is it a geometric or telescoping (only ones we have studied so far)? Feb 25 12:01 PM 32

33 Feb 25 12:05 PM 33

34 Feb 25 12:05 PM 34

35 Feb 25 12:05 PM 35

36 Feb 25 1:09 PM 36

37 Feb 25 1:28 PM 37

38 Feb 25 1:32 PM 38

39 P- Series Can be written in the form Characteristics: * notice that the "counter" is the base and "p" is the power Conditions: * * **will be helpful with the comparison test later Feb 24 3:42 PM 39

40 ex 1 ex 2 Feb 24 7:29 PM 40

41 a special case of the p-series harmonic series Note: the harmonic sequence converges but the series diverges Why? Feb 24 7:33 PM 41

42 Warm ups: From note packet Feb 26 12:09 PM 42

43 Alternating- Series Can be written in the form given and/or converges if : means that the series is decreasing Feb 24 7:38 PM 43

44 Feb 26 12:20 PM 44

45 separate ( 1) does lim = 0 is it decreasing? Feb 25 1:39 PM 45

46 Feb 25 1:33 PM 46

47 Feb 26 12:10 PM 47

48 Feb 26 12:10 PM 48

49 nth divergence test if no diverges if yes continue "known" series geometric? telescoping p series alternating Feb 26 12:12 PM 49

50 Once you've worked all of the ones that you recognize it's time to "test" the remaining series Integral test If f(x) is a positive, decreasing and continuous function on such that then means that when n is the input, f(n) generates the terms of the series behave the same way Feb 24 7:58 PM 50

51 Likewise Feb 24 8:09 PM 51

52 ex 1 Let's use the harmonic series although we already know how it behaves (remember?) nth divergence test first means it's possible to converge conditions to use integral test: positive, decreasing and cont Feb 24 8:15 PM 52

53 A Visual look at why the Integral test works Jan 17 8:03 AM 53

54 Jan 17 8:03 AM 54

55 Determine if the given infinite series converges or diverges Jan 17 8:04 AM 55

56 Jan 17 9:04 AM 56

57 ex 2 Game plan 1. check if possible to converge 2. check conditions to use integral test 3. rewrite as f(x) and use integral from 1 to inf Feb 24 8:24 PM 57

58 ex 2 Game plan 1. check if possible to converge 2. check conditions to use integral test 3. rewrite as f(x) and use integral from 1 to inf Feb 24 8:24 PM 58

59 Feb 25 12:49 PM 59

60 Comparison test (also called direct comparison test) means that each term of a n is less than or equal to each term of b n if "big" converges "little" is trapped beneath and converges if "little" diverges each term of "big" is larger so it also diverges Feb 24 8:27 PM 60

61 Feb 26 12:21 PM 61

62 ex 1 game plan: 1) show that it fits the conditions 2) need to compare it to a series that behaves similarly that we already know how it behaves Feb 24 8:39 PM 62

63 ex 2 game plan: 1) show that it fits the conditions 2) need to compare it to a series that behaves similarly that we already know how it behaves Feb 24 8:42 PM 63

64 Feb 26 12:24 PM 64

65 Mar 2 12:32 PM 65

66 Ratio test helpful for series involving exponentials and factorials is finite conditions: means a n+1 < a n decreasing means a n < a n+1 inc Feb 24 8:45 PM 66

67 Feb 26 12:23 PM 67

68 ex 1 Feb 24 8:45 PM 68

69 ex 1 Feb 24 8:45 PM 69

70 ex 2 Feb 24 9:06 PM 70

71 ex. 3 c Feb 24 9:07 PM 71

72 From notes Mar 2 12:16 PM 72

73 Root test helpful when series is raised to the nth power Feb 24 8:45 PM 73

74 ex 1 c Feb 24 8:45 PM 74

75 ex. 2 d Feb 24 9:16 PM 75

76 Mar 3 1:03 PM 76

77 ex. 3 Feb 24 9:19 PM 77

78 Mar 2 12:16 PM 78

79 Homework Series convergent or divergent? Feb 27 2:01 PM 79