Sequences and Series
|
|
- Denis Lynch
- 5 years ago
- Views:
Transcription
1 Sequences and Series What do you think of when you read the title of our next unit? In case your answers are leading us off track, let's review the following IB problems. 1
2 November 2013 HL 2
3 3
4 November 2008 HL 4
5 5
6 May 2013 HL 6
7 7
8 Some history It's this last type, the infinite series that had puzzled mathematicians for centuries, especially the summation of an infinite series. Why do some infinite series add to a number??? Such as: This can be seen by adding the areas in the "infinitely halved" unit square below
9 But sometimes the infinite sum is infinite! Such as: (don't worry we will prove this beyond a reasonable doubt). 9
10 And sometimes the infinite sum is impossible to pin down! Such as: = Is it 1? Is it 0? Is it neither? What an exciting time to be a mathematician...or a BC Calculus student! By the end of this unit you will have developed an intuitive sense about infinite series and learn tests for convergence. 10
11 Misconception An infinite series is not an example of addition. Addition of real numbers is a binary operation, adding numbers two at a time. A finite sum of real numbers always produces a real number because it is a finite number of binary additions. An infinite sum of real numbers is something else entirely...which leads us to the following definition. An infinite series is an expression of the form a 1 + a 2 + a a n +..., or. The numbers a 1, a 2, a 3,... are the terms of the series; a n is the n th term. 11
12 The partial sums of the series form a sequence S 1 = a 1 S 2 = a 1 + a 2 S 3 = a 1 + a 2 + a 3 S n = of real numbers, each defined as a finite sum. If the sequence of partial sums has a limit L as n, we say that the series converges to the sum L and = L. Otherwise, we say that it diverges. 12
13 Example 1: Does the series converge? 13
14 Example 2: Does the series converge? 14
15 There is an easy way to identify some divergent series. If the infinite series converges, then This means that if the series must diverge. This is referred to as the n th term test. Does it work? Let's revisit Example 2: 15
16 16
17 Do all infinite geometric series converge? 17
18 Example 3: Tell whether each series converges or diverges. If it converges, give its sum. (a) (b) (c) (d) 18
19 How are we feeling about series thus far? Confused Happy Stunned We have just started our journey into infinite series but we know a lot about the convergence and divergence of an entire class of series (geometric series). Let's continue our journey by incorporating x. 19
20 Representing Functions by Series Let's stick with infinite geometric series for a little bit. What can you conclude about the series below? vs. So what can be said about the following series? 20
21 Graphic Representation 21
22 Power Series An expression of the form is a power series centered at x = 0. An expression of the form is a power series centered at x = a. The term center. is the n th term; the number a is the An example of power series, you ask? What about our geometric series? This is a power series centered at x =. It converges on the interval This interval is also centered at x =. This is not a coincidence. This is typical behavior for a power series. A power series will either: 1) converge for all x, 2) converges on a finite interval with the same center as the series, or 3) converges only at the center itself. 22
23 We have seen that the power series represents the function on the domain ( 1, 1). Can we find power series to represent other functions? Exploration 1: Finding Power Series for Other Functions Given that is represented by the power series on the interval ( 1, 1), 1. express as a power series and find its interval of convergence. 2. express as a power series and find its interval of convergence. 23
24 3. express as a power series and find its interval of convergence. 4. express as a power series and find its interval of convergence. 24
25 Finding a Power Series by Differentiation So far we have only represented functions by power series that happen to be geometric. The partial sums that converge to those power series are polynomials and we can apply calculus to polynomials so it would appear that the calculus of polynomials would also apply to power series. Example 4: Given that is represented by the power series on the interval ( 1, 1), find a power series to represent. 25
26 Theorem: Term by Term Differentiation and Integration If converges for some interval of convergence, then the series obtained by differentiating the series for f(x) term by term f '(x) = converges for the same interval of convergence and represents f '(x) on that interval. If the series for f(x) converges for all x, then so does the series for f '(x). 26
27 Example 5: Finding a Power Series by Integration 27
28 Theorem: Term by Term Integration If converges for, then the series obtained by integrating the series for f(x) term by term converges for and represents on that interval. If the series for f(x) converges for all x, then so does the series for the integral. 28
29 Example 6: Given that find a power series to represent ln(1 + x). 29
30 Example 7: 30
31 Identifying a Series So far we have been finding power series to represent functions. Let us try to find the function that a given power series represents. Exploration: Define a function f by a power series as follows: 1. Find f '(x). 2. Find f(0). 3. What well known function do you suppose f(x) is? 4. Graph the first three partial sums. What appears to be the interval of convergence? 5. Graph the next three partial sums. Did you underestimate the interval of convergence? 31
32 Taylor Series Here we are going to learn a more general technique for constructing a power series. First, let's start by constructing a polynomial. Exploration: Construct a polynomial with the following behavior at x = 0: 32
33 Exploration 2: Let's see if our general form for the coefficient holds up!!! 33
34 34
35 Example 1: Construct a polynomial that matches the behavior of ln(x + 1) at x = 0 through its first four derivatives. 35
36 We have just constructed the fourth order Taylor polynomial for the function ln(1 + x) at x = 0. Does it look familiar? 36
37 We can use this technique to construct Taylor Series about x = 0 for any function, as long as we can keep taking derivatives there. Two functions that are particularly well suited for this are sine and cosine. Example 2: Construct the seventh order Taylor polynomial and the Taylor Series for sin x at x = 0. 37
38 Now let's graph the first nine partial sums together with y = sin x to see how well we did. 38
39 Example 3: Construct the sixth order Taylor polynomial and the Taylor series at x = 0 for cos x. 39
40 Definition: Taylor Series Generated by f at x = 0 (Maclaurin Series) Let f be a function with derivatives of all orders throughout some open interval containing 0. Then the Taylor series generated by f at x = 0 is This series is also called the Maclaurin series generated by f. The partial sum is the Taylor polynomial of order n for f at x = 0. 40
41 Example 3.5: Find the Maclaurin Series for a previous exploration?. Does it resemble 41
42 Example 4: Find the fourth order Taylor polynomial that approximates y = cos(2x) near x = 0. 42
43 Most Useful Maclaurin Series 43
44 But when are we going to use this??? These polynomial approximations are useful in a variety of ways. First, it is easy to do calculus with polynomials. Second, polynomials are built using only two basic operations (addition and multiplication), so computers can handle them easily. However sometimes being restricted to a power series at x = 0 can be limiting...but why are we limited? We aren't! We can match a power series with f in the same way at any value x = a, provided we can take the derivatives. We get this formula simply by "shifting horizontally". Definition: Taylor Series Generated by f at x = a Let f be a function with derivatives of all orders throughout some open interval containing a. Then the Taylor series generated by f at x = a is The partial sum is the Taylor polynomial of order n for f at x = a. 44
45 Example 5: Find the Taylor series generated by f(x) = e x at x = 2. 45
46 Example 6: Find the third order Taylor polynomial for f(x) = 2x 3 3x 2 + 4x 5 (a) at x = 0 (b) at x = 1. 46
47 How can we find the Power Series of a function from the known Power Series of another function? 1) Find the Maclaurin series for f(x) = cos x by differentiating the Maclaurin series for sin x. 47
48 2) Using the Maclaurin series for cos x, find the Maclaurin series for. 48
49 3) Using the Maclaurin series for, find the Maclaurin series for. 49
50 4) (a) Using the Maclaurin series for find the Maclaurin series for. (b) Find the Maclaurin series for h such that and h(0) = 5. 50
51 Putting Taylor Polynomials to Use It is nice to know that sin(x) can be found exactly by summing an infinite Taylor series, but if we want to use that information to find sin(3), we will have to evaluate Taylor polynomials until we arrive at an approximation that we are satisfied with. Even our computers and calculators have to deal with this! 51
52 Example 1: Find a Taylor polynomial that will serve as an adequate substitute for sin(x) on the interval [ π, π]. 52
53 We want to be able to use Taylor polynomials to approximate functions over the intervals of convergence of the Taylor series and we would like to keep the error of the approximation within specified bounds. Since the error results from truncating the series down to a polynomial (cutting it off after some number of terms), we call it the truncation error. 53
54 Example 2: Find a formula for the truncation error if we use 1 + x 2 + x 4 + x 6 to approximate the interval ( 1, 1). over 54
55 How can we handle the error if we were to truncate a nongeometric series? Taylor's Theorem Every truncation splits a Taylor series into two equally significant pieces; 1) the Taylor polynomial P n (x) this gives the approximation 2) and the remainder R n (x) this tells us whether the approximation is any good Taylor's theorem is about both pieces. 55
56 Taylor's Theorem with Remainder If f has derivatives of all orders in an open interval I containing a, then for each positive integer n and for each x in I, where for some c between a and x. Taylor's formula: Remainder of order n: also called the LaGrange form of the remainder 56
57 Taylor's Theorem is a pretty amazing result. Breaking it down, it says that a function with (n+1) derivatives can be evaluated EXACTLY at a point by a degree (n + 1) polynomial AND tells us exactly how to construct that polynomial. Wow!!! Insert shocked face! Or possibly confused face... 57
58 Let's do an example to illustrate Taylor's Theorem. Consider expanding e x about x = 0 out to a third degree polynomial. P 3 (x) = Let's use this polynomial to approximate e 2. We get e 2 Taylor's Theorem guarantees the existence of a number c, with 0 < c < 2 such that e 2 = P 3 (x) + R 3 (x) = 58
59 Let's visualize what is going on. Let's graph f(x) = e x along with P 3 (x). If we use the polynomial to approximate e 2, the error is the length of the short vertical segment from f(x) to P 3 (x). 59
60 Let's find the length of that vertical segment. Then we can use that to find c. 60
61 Do we really need to find the value of c? Could we tell anything by just looking at the behavior of R n (x) overall? Example 3: Prove that the series converges to sin(x) for all real x. 61
62 Example 4: Prove that the series converges to cos(x) for all real x. 62
63 What are the important consequences of Taylor's Theorem? 1) We saw that we never really care to find the actual value of c (we did it just this once to prove to you that there really will be such a c that exists). Instead we would rather determine some maximum possible value that the magnitude of our remainder term could be for a particular n. Meaning how big could the absolute value of the (n + 1) st derivative of our function be, assuming c is somewhere between a and x? If we can find a bound for the absolute value of that derivative, we can find a bound for the absolute value of the remainder term and a bound on the error. This is called the LaGrange error bound. 63
64 Remainder Estimation Theorem We were able to use the remainder formula in Taylor's Theorem to verify the convergence of two Taylor series to their generating functions (sin(x) and cos(x)) and we did NOT need an actual value for. Instead we were able to put an upper bound on for all x. which was enough to ensure that Remainder Estimation Theorem If the function f can be differentiated (n+1) times on an interval I containing the number a and if M is an upper bound for on I, meaning for all x in I, then for all x in I. 64
65 Example 5: The approximation is used when x is small. Use the Remainder Estimation Theorem to get a bound for the maximum error when. 65
66 What are the important consequences of Taylor's Theorem? 2) If we can determine that the remainder term goes to 0 as n, then we know the Taylor series converges to the function value at the point in question. What's cool is that as long as the (n + 1) st derivative is bounded on the interval in question and independent of n, the remainder term will go to 0 because of GRANNY FEPL! Factorials grow faster than power functions. 66
67 67
68 68
69 69
70 70
71 71
72 72
73 73
74 74
75 75
76 76
77 Consider the following math sentence: Is this always true for all values of x? 77
78 So far we have had no issue discussing convergence for geometric series and even some of our power series, thanks to Taylor's Theorem. But we need more... 78
79 The Convergence Theorem for Power Series There are three possibilities for with respect to convergence. 1) There is a positive number R such that the series diverges for but converges for. The series may or may not converge at either of the endpoints x = a R and x = a + R. 2) The series converges for every x. (R = ) 3) The series converges at x = a and diverges elsewhere. (R = 0) R = radius of convergence 79
80 Test 1: n th Term Test for Divergence The most obvious requirement for convergence of a series is that the n th term must go to zero as n approaches infinity. diverges if fails to exist or is different from zero. Examples: Using the n th Term Test for Divergence, what conclusion can you draw about the convergence or divergence of the given series? 1) 2) 3) 4) 80
81 Test 2: The Direct Comparison Test Let be a series with no negative terms. (a) converges if there is a convergent series with a n c n for all n > N, for some integer N. (b) diverges if there is a divergent series of nonnegative terms with a n d n for all n > N, for some integer N. 81
82 Examples: Use the Direct Comparison Test to determine whether the series converges or diverges. 1) 2) 3) 82
83 Definition: Absolute Convergence If the series of absolute values converges, then converges absolutely. Theorem: If converges, then converges. We use this when the series has some positive and negative terms but it doesn't alternate from positive to negative. More on that later! Example: Show that converges for all x. 83
84 84
85 85
86 86
87 Examples: 1) 2) 87
88 Test 3: The Ratio Test Let be a series with. Then, (a) the series converges if L < 1, (b) the series diverges if L > 1, (c) the test is inconclusive if L = 1 and you must perform another test. We also use the Ratio Test to find the Interval of Convergence of a series. 88
89 Examples: Use the Ratio Test to determine whether the series converges or diverges. 1) 2) 89
90 Examples: Use the Ratio Test to determine whether the series converges or diverges. 3) 90
91 Endpoint Convergence The Ratio Test, which is really a test for absolute convergence, establishes the radius of convergence for. But then this is the same as the radius of convergence of So all that remains to be resolved about the convergence of an arbitrary power series is the question of convergence at the endpoints of the convergence when the radius of convergence is a finite, nonzero number. Example: Find the interval of convergence of 91
92 Example: Find the interval of convergence for each power series. 2) 92
93 Test 4: The Integral Test Let {a n } be a sequence of positive terms. Suppose that a n = f(n), where f is a continuous, positive, decreasing function of x for all x N (N is a positive integer). Then the series and the integral either both converge or diverge. Note: If the integral exists then it just means the series converges, the value of the integral is not the value of the series. 93
94 Example: Does converge? 94
95 The Integral Test can be used to settle the question of convergence for any series of the form:, p is a real constant. We call these series a p series. 95
96 Exploration: The p Series Test 1) Use the Integral Test to prove that converges if p > 1. 2) Use the Integral Test to prove that diverges if p < 1. 3) Use the Integral Test to prove that diverges if p = 1. 96
97 The p Series Test If p 1, then the p series diverges. If p > 1, then the p series converges. Note: There is no formula to determine the value to which the series converges. 97
98 Examples: List the first 3 terms of each series, state the value of p, and state if the series converges or diverges. 1) 2) 3) 98
99 The Harmonic Series This is the most famous divergent series in mathematics. The p Series Test shows that the harmonic series is just barely divergent; if we increase p to , the series converges! It is difficult to see by just adding up the terms because the partial sums increase at a slower and slower rate. For example, the sum of the first 4 terms edges over 2; 11 terms edges over 3; 227 terms edges over 6; 12,367 terms edges over 10; and about 250 million terms edges over 20. Proof that the harmonic series diverges: Group the terms of the harmonic series as follows: Compare this to the series: or Notice that each term in each grouping of the harmonic series is greater than or equal to each corresponding term in the corresponding grouping of the second series. Since the second series, the harmonic series must also. 99
100 Example: Find the interval of convergence for the power series. 100
101 Test 5: The Limit Comparison Test Suppose that a n > 0 and b n > 0 for all n N (N a positive integer) 1) If then and both converge or diverge. 2) If and converges, then converges. 3) If and diverges, then diverges. Example: To show that the series diverges, we can compare this to which is the divergent harmonic series. 101
102 The Limit Comparison Test works well for comparing "messy" algebraic series to a p series. When choosing an appropriate series to which we want to make a comparison, we disregard all but the highest powers of n in both the numerator and the denominator. Example: 1) Given, choose 2) Given, choose 3) Given, choose 4) Given, choose 102
103 Example: Determine whether the series converge or diverge, a) b) c) d) 103
104 What is an Alternating Series? An alternating series is a series whose terms alternate in sign. For example: Alternating Series occur in two ways: 104
105 Alternating Series Test If a n > 0, then the alternating series and converge, provided both of the following conditions are met: 1) 2) 105
106 Examples: 1) 2) 3) 106
107 Illustrating the convergence of the partial sums to their limit L. (Closing in on the sum of a convergent alternating series) 0 S 2 S 4 L S 3 S 1 107
108 Alternate Series Remainder Theorem Examples: 1) Approximate the sum and error of the series from its first six terms. 108
109 Example: Prove that the alternating harmonic series is convergent, but not absolutely convergent. Find a bound for the truncation error after 99 terms. 109
110 Absolute and Conditional Convergence A series is absolutely convergent if A series is conditionally convergent if 110
111 Examples: 1) 2) 111
112 Testing for Convergence/Divergence: Which Test to Use The following are guidelines, not rules. 1) Always first see if the terms go to zero. If they don't, then the series diverges (n th term test). If the terms do go to zero, no conclusion can be drawn; try another test. 2) Use the direct comparison test if the series looks like one you know, such as the harmonic series, p series, or geometric series. 3) If you can't use the direct comparison test, but the series looks like one you know, try the limit comparison test. Also, the limit comparison test may be tried if the series is a fraction whose numerator and denominator are polynomials or roots of polynomials. 4) Use the integral test if the series looks like an integral you have done. Don't use the integral test if you can see an easier one to use. 112
113 Testing for Convergence/Divergence: Which Test to Use 5) Use the ratio test if the series contains combinations of variables in exponents, variables in bases or factorials. 6) Use the root test if the series is an expression to the n th power. 7) If the series has alternating signs (it will have a factor such as ( 1) n or ( 1) n+1 in it), use the alternating series test. 8) If the series contains positive and negative terms (not necessarily alternating), you may also test to see if the series converges absolutely. If it does, then the original series converges. 9) Remember Dominance: n n > Factorials > Exponentials > Powers > Logs x x > x! > b x (b > 1) > x c (c > 0) > ln x Granny FEPL 113
114 114
115 115
116 116
117 117
118 118
119 119
120 120
121 121
122 122
123 123
124 124
125 125
126 126
127 127
128 128
129 129
130 130
131 131
132 132
133 133
134 134
135 135
136 136
137 137
8.5 Taylor Polynomials and Taylor Series
8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES 50 8.5 Taylor Polynomials and Taylor Series Motivating Questions In this section, we strive to understand the ideas generated by the following important questions:
More informationLet s Get Series(ous)
Department of Mathematics, Computer Science, and Statistics Bloomsburg University Bloomsburg, Pennsylvania 785 Let s Get Series(ous) Summary Presenting infinite series can be (used to be) a tedious and
More informationChapter 11 - Sequences and Series
Calculus and Analytic Geometry II Chapter - Sequences and Series. Sequences Definition. A sequence is a list of numbers written in a definite order, We call a n the general term of the sequence. {a, a
More informationAP Calculus Chapter 9: Infinite Series
AP Calculus Chapter 9: Infinite Series 9. Sequences a, a 2, a 3, a 4, a 5,... Sequence: A function whose domain is the set of positive integers n = 2 3 4 a n = a a 2 a 3 a 4 terms of the sequence Begin
More informationSUMMATION TECHNIQUES
SUMMATION TECHNIQUES MATH 53, SECTION 55 (VIPUL NAIK) Corresponding material in the book: Scattered around, but the most cutting-edge parts are in Sections 2.8 and 2.9. What students should definitely
More informationMATH 1231 MATHEMATICS 1B CALCULUS. Section 5: - Power Series and Taylor Series.
MATH 1231 MATHEMATICS 1B CALCULUS. Section 5: - Power Series and Taylor Series. The objective of this section is to become familiar with the theory and application of power series and Taylor series. By
More informationInfinite Series. Copyright Cengage Learning. All rights reserved.
Infinite Series Copyright Cengage Learning. All rights reserved. Taylor and Maclaurin Series Copyright Cengage Learning. All rights reserved. Objectives Find a Taylor or Maclaurin series for a function.
More informationFebruary 13, Option 9 Overview. Mind Map
Option 9 Overview Mind Map Return tests - will discuss Wed..1.1 J.1: #1def,2,3,6,7 (Sequences) 1. Develop and understand basic ideas about sequences. J.2: #1,3,4,6 (Monotonic convergence) A quick review:
More informationMath WW09 Solutions November 24, 2008
Math 352- WW09 Solutions November 24, 2008 Assigned problems: 8.7 0, 6, ww 4; 8.8 32, ww 5, ww 6 Always read through the solution sets even if your answer was correct. Note that like many of the integrals
More informationMATH 1231 MATHEMATICS 1B CALCULUS. Section 4: - Convergence of Series.
MATH 23 MATHEMATICS B CALCULUS. Section 4: - Convergence of Series. The objective of this section is to get acquainted with the theory and application of series. By the end of this section students will
More informationMath 113 (Calculus 2) Exam 4
Math 3 (Calculus ) Exam 4 November 0 November, 009 Sections 0, 3 7 Name Student ID Section Instructor In some cases a series may be seen to converge or diverge for more than one reason. For such problems
More informationMITOCW watch?v=y6ma-zn4olk
MITOCW watch?v=y6ma-zn4olk PROFESSOR: We have to ask what happens here? This series for h of u doesn't seem to stop. You go a 0, a 2, a 4. Well, it could go on forever. And what would happen if it goes
More information19. TAYLOR SERIES AND TECHNIQUES
19. TAYLOR SERIES AND TECHNIQUES Taylor polynomials can be generated for a given function through a certain linear combination of its derivatives. The idea is that we can approximate a function by a polynomial,
More information8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0
8.7 Taylor s Inequality Math 00 Section 005 Calculus II Name: ANSWER KEY Taylor s Inequality: If f (n+) is continuous and f (n+) < M between the center a and some point x, then f(x) T n (x) M x a n+ (n
More informationMATH141: Calculus II Exam #4 review solutions 7/20/2017 Page 1
MATH4: Calculus II Exam #4 review solutions 7/0/07 Page. The limaçon r = + sin θ came up on Quiz. Find the area inside the loop of it. Solution. The loop is the section of the graph in between its two
More informationMath 106: Review for Final Exam, Part II - SOLUTIONS. (x x 0 ) 2 = !
Math 06: Review for Final Exam, Part II - SOLUTIONS. Use a second-degree Taylor polynomial to estimate 8. We choose f(x) x and x 0 7 because 7 is the perfect cube closest to 8. f(x) x /3 f(7) 3 f (x) 3
More informationChapter 4 Sequences and Series
Chapter 4 Sequences and Series 4.1 Sequence Review Sequence: a set of elements (numbers or letters or a combination of both). The elements of the set all follow the same rule (logical progression). The
More informationCHALLENGE! (0) = 5. Construct a polynomial with the following behavior at x = 0:
TAYLOR SERIES Construct a polynomial with the following behavior at x = 0: CHALLENGE! P( x) = a + ax+ ax + ax + ax 2 3 4 0 1 2 3 4 P(0) = 1 P (0) = 2 P (0) = 3 P (0) = 4 P (4) (0) = 5 Sounds hard right?
More informationSequences and infinite series
Sequences and infinite series D. DeTurck University of Pennsylvania March 29, 208 D. DeTurck Math 04 002 208A: Sequence and series / 54 Sequences The lists of numbers you generate using a numerical method
More informationPower series and Taylor series
Power series and Taylor series D. DeTurck University of Pennsylvania March 29, 2018 D. DeTurck Math 104 002 2018A: Series 1 / 42 Series First... a review of what we have done so far: 1 We examined series
More information6: Polynomials and Polynomial Functions
6: Polynomials and Polynomial Functions 6-1: Polynomial Functions Okay you know what a variable is A term is a product of constants and powers of variables (for example: x ; 5xy ) For now, let's restrict
More information11.10a Taylor and Maclaurin Series
11.10a 1 11.10a Taylor and Maclaurin Series Let y = f(x) be a differentiable function at x = a. In first semester calculus we saw that (1) f(x) f(a)+f (a)(x a), for all x near a The right-hand side of
More informationMath Numerical Analysis
Math 541 - Numerical Analysis Joseph M. Mahaffy, jmahaffy@mail.sdsu.edu Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State University
More informationMath 1b Sequences and series summary
Math b Sequences and series summary December 22, 2005 Sequences (Stewart p. 557) Notations for a sequence: or a, a 2, a 3,..., a n,... {a n }. The numbers a n are called the terms of the sequence.. Limit
More informationMain topics for the First Midterm Exam
Main topics for the First Midterm Exam The final will cover Sections.-.0, 2.-2.5, and 4.. This is roughly the material from first three homeworks and three quizzes, in addition to the lecture on Monday,
More information3.4 Introduction to power series
3.4 Introduction to power series Definition 3.4.. A polynomial in the variable x is an expression of the form n a i x i = a 0 + a x + a 2 x 2 + + a n x n + a n x n i=0 or a n x n + a n x n + + a 2 x 2
More informationMITOCW ocw f07-lec39_300k
MITOCW ocw-18-01-f07-lec39_300k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.
More informationn=1 ( 2 3 )n (a n ) converges by direct comparison to
. (a) n = a n converges, so we know that a n =. Therefore, for n large enough we know that a n
More information1 Question related to polynomials
07-08 MATH00J Lecture 6: Taylor Series Charles Li Warning: Skip the material involving the estimation of error term Reference: APEX Calculus This lecture introduced Taylor Polynomial and Taylor Series
More informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More informationA sequence { a n } converges if a n = finite number. Otherwise, { a n }
9.1 Infinite Sequences Ex 1: Write the first four terms and determine if the sequence { a n } converges or diverges given a n =(2n) 1 /2n A sequence { a n } converges if a n = finite number. Otherwise,
More informationConceptual Explanations: Simultaneous Equations Distance, rate, and time
Conceptual Explanations: Simultaneous Equations Distance, rate, and time If you travel 30 miles per hour for 4 hours, how far do you go? A little common sense will tell you that the answer is 120 miles.
More informationMath101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:
Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the
More informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
More informationTaylor and Maclaurin Series. Copyright Cengage Learning. All rights reserved.
11.10 Taylor and Maclaurin Series Copyright Cengage Learning. All rights reserved. We start by supposing that f is any function that can be represented by a power series f(x)= c 0 +c 1 (x a)+c 2 (x a)
More information8.7 MacLaurin Polynomials
8.7 maclaurin polynomials 67 8.7 MacLaurin Polynomials In this chapter you have learned to find antiderivatives of a wide variety of elementary functions, but many more such functions fail to have an antiderivative
More information2t t dt.. So the distance is (t2 +6) 3/2
Math 8, Solutions to Review for the Final Exam Question : The distance is 5 t t + dt To work that out, integrate by parts with u t +, so that t dt du The integral is t t + dt u du u 3/ (t +) 3/ So the
More informationCalculus Favorite: Stirling s Approximation, Approximately
Calculus Favorite: Stirling s Approximation, Approximately Robert Sachs Department of Mathematical Sciences George Mason University Fairfax, Virginia 22030 rsachs@gmu.edu August 6, 2011 Introduction Stirling
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson
JUST THE MATHS UNIT NUMBER.5 DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) by A.J.Hobson.5. Maclaurin s series.5. Standard series.5.3 Taylor s series.5.4 Exercises.5.5 Answers to exercises
More informationM155 Exam 2 Concept Review
M155 Exam 2 Concept Review Mark Blumstein DERIVATIVES Product Rule Used to take the derivative of a product of two functions u and v. u v + uv Quotient Rule Used to take a derivative of the quotient of
More informationTopic 5 Notes Jeremy Orloff. 5 Homogeneous, linear, constant coefficient differential equations
Topic 5 Notes Jeremy Orloff 5 Homogeneous, linear, constant coefficient differential equations 5.1 Goals 1. Be able to solve homogeneous constant coefficient linear differential equations using the method
More informationLecture 5: Function Approximation: Taylor Series
1 / 10 Lecture 5: Function Approximation: Taylor Series MAR514 Geoffrey Cowles Department of Fisheries Oceanography School for Marine Science and Technology University of Massachusetts-Dartmouth Better
More informationEverything Old Is New Again: Connecting Calculus To Algebra Andrew Freda
Everything Old Is New Again: Connecting Calculus To Algebra Andrew Freda (afreda@deerfield.edu) ) Limits a) Newton s Idea of a Limit Perhaps it may be objected, that there is no ultimate proportion of
More informationSequences and Series
Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite
More informationConvergence of sequences and series
Convergence of sequences and series A sequence f is a map from N the positive integers to a set. We often write the map outputs as f n rather than f(n). Often we just list the outputs in order and leave
More informationTaylor Series. richard/math230 These notes are taken from Calculus Vol I, by Tom M. Apostol,
Taylor Series Professor Richard Blecksmith richard@math.niu.edu Dept. of Mathematical Sciences Northern Illinois University http://math.niu.edu/ richard/math230 These notes are taken from Calculus Vol
More informationHUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK
HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK COURSE / SUBJECT A P C a l c u l u s ( B C ) KEY COURSE OBJECTIVES/ENDURING UNDERSTANDINGS OVERARCHING/ESSENTIAL SKILLS OR QUESTIONS Limits and Continuity Derivatives
More informationMath 0230 Calculus 2 Lectures
Math 00 Calculus Lectures Chapter 8 Series Numeration of sections corresponds to the text James Stewart, Essential Calculus, Early Transcendentals, Second edition. Section 8. Sequences A sequence is a
More informationSequences and Series. 256 Chapter 11 Sequences and Series. and then lim 1 1 = 1 0 = 1.
256 Chapter Sequences and Series Consider the following sum: Sequences and Series 2 + 4 + 8 + 6 + + 2 + i The dots at the end indicate that the sum goes on forever Does this make sense? Can we assign a
More informationc n (x a) n c 0 c 1 (x a) c 2 (x a) 2...
3 CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS 6. REVIEW OF POWER SERIES REVIEW MATERIAL Infinite series of constants, p-series, harmonic series, alternating harmonic series, geometric series, tests
More informationbase 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation.
EXPONENTIALS Exponential is a number written with an exponent. The rules for exponents make computing with very large or very small numbers easier. Students will come across exponentials in geometric sequences
More information1 + lim. n n+1. f(x) = x + 1, x 1. and we check that f is increasing, instead. Using the quotient rule, we easily find that. 1 (x + 1) 1 x (x + 1) 2 =
Chapter 5 Sequences and series 5. Sequences Definition 5. (Sequence). A sequence is a function which is defined on the set N of natural numbers. Since such a function is uniquely determined by its values
More informationTAYLOR POLYNOMIALS DARYL DEFORD
TAYLOR POLYNOMIALS DARYL DEFORD 1. Introduction We have seen in class that Taylor polynomials provide us with a valuable tool for approximating many different types of functions. However, in order to really
More informationLast/Family Name First/Given Name Seat #
Math 2, Fall 27 Schaeffer/Kemeny Final Exam (December th, 27) Last/Family Name First/Given Name Seat # Failure to follow the instructions below will constitute a breach of the Stanford Honor Code: You
More informationReview Sheet on Convergence of Series MATH 141H
Review Sheet on Convergence of Series MATH 4H Jonathan Rosenberg November 27, 2006 There are many tests for convergence of series, and frequently it can been confusing. How do you tell what test to use?
More informationA Hypothesis about Infinite Series, of the minimally centered variety. By Sidharth Ghoshal May 17, 2012
A Hypothesis about Infinite Series, of the minimally centered variety By Sidharth Ghoshal May 17, 2012 Contents Taylor s Theorem and Maclaurin s Special Case: A Brief Introduction... 3 The Curious Case
More informationSolving Quadratic & Higher Degree Equations
Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationSolved problems: (Power) series 1. Sum up the series (if it converges) 3 k+1 a) 2 2k+5 ; b) 1. k(k + 1).
Power series: Solved problems c phabala 00 3 Solved problems: Power series. Sum up the series if it converges 3 + a +5 ; b +.. Investigate convergence of the series a e ; c ; b 3 +! ; d a, where a 3. Investigate
More informationAP Calculus BC Scope & Sequence
AP Calculus BC Scope & Sequence Grading Period Unit Title Learning Targets Throughout the School Year First Grading Period *Apply mathematics to problems in everyday life *Use a problem-solving model that
More information(c) Find the equation of the degree 3 polynomial that has the same y-value, slope, curvature, and third derivative as ln(x + 1) at x = 0.
Chapter 7 Challenge problems Example. (a) Find the equation of the tangent line for ln(x + ) at x = 0. (b) Find the equation of the parabola that is tangent to ln(x + ) at x = 0 (i.e. the parabola has
More informationMath 106 Fall 2014 Exam 2.1 October 31, ln(x) x 3 dx = 1. 2 x 2 ln(x) + = 1 2 x 2 ln(x) + 1. = 1 2 x 2 ln(x) 1 4 x 2 + C
Math 6 Fall 4 Exam. October 3, 4. The following questions have to do with the integral (a) Evaluate dx. Use integration by parts (x 3 dx = ) ( dx = ) x3 x dx = x x () dx = x + x x dx = x + x 3 dx dx =
More informationInfinite series, improper integrals, and Taylor series
Chapter 2 Infinite series, improper integrals, and Taylor series 2. Introduction to series In studying calculus, we have explored a variety of functions. Among the most basic are polynomials, i.e. functions
More informationMATH 118, LECTURES 27 & 28: TAYLOR SERIES
MATH 8, LECTURES 7 & 8: TAYLOR SERIES Taylor Series Suppose we know that the power series a n (x c) n converges on some interval c R < x < c + R to the function f(x). That is to say, we have f(x) = a 0
More informationMATH 13100/58 Class 6
MATH 13100/58 Class 6 Minh-Tam Trinh Today and Friday, we cover the equivalent of Chapters 1.1-1.2 in Purcell Varberg Rigdon. 1. Calculus is about two kinds of operations on functions: differentiation
More informationERRATA for Calculus: The Language of Change
1 ERRATA for Calculus: The Language of Change SECTION 1.1 Derivatives P877 Exercise 9b: The answer should be c (d) = 0.5 cups per day for 9 d 0. SECTION 1.2 Integrals P8 Exercise 9d: change to B (11) P9
More informationSection 11.1: Sequences
Section 11.1: Sequences In this section, we shall study something of which is conceptually simple mathematically, but has far reaching results in so many different areas of mathematics - sequences. 1.
More information2. Limits at Infinity
2 Limits at Infinity To understand sequences and series fully, we will need to have a better understanding of its at infinity We begin with a few examples to motivate our discussion EXAMPLE 1 Find SOLUTION
More informationAlgebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Functions
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 2017/2018 DR. ANTHONY BROWN 4. Functions 4.1. What is a Function: Domain, Codomain and Rule. In the course so far, we
More informationMa 530 Power Series II
Ma 530 Power Series II Please note that there is material on power series at Visual Calculus. Some of this material was used as part of the presentation of the topics that follow. Operations on Power Series
More informationCHAPTER 1 Prerequisites for Calculus 2. CHAPTER 2 Limits and Continuity 58
CHAPTER 1 Prerequisites for Calculus 2 1.1 Lines 3 Increments Slope of a Line Parallel and Perpendicular Lines Equations of Lines Applications 1.2 Functions and Graphs 12 Functions Domains and Ranges Viewing
More informationV. Graph Sketching and Max-Min Problems
V. Graph Sketching and Max-Min Problems The signs of the first and second derivatives of a function tell us something about the shape of its graph. In this chapter we learn how to find that information.
More informationSign of derivative test: when does a function decreases or increases:
Sign of derivative test: when does a function decreases or increases: If for all then is increasing on. If for all then is decreasing on. If then the function is not increasing or decreasing at. We defined
More informationMATH 1231 MATHEMATICS 1B Calculus Section 4.3: - Series.
MATH 1231 MATHEMATICS 1B 2010. Calculus Section 4.3: - Series. 1. Sigma notation 2. What is a series? 3. The big question 4. What you should already know 5. Telescoping series 5. Convergence 6. n th term
More informationHigh School AP Calculus BC Curriculum
High School AP Calculus BC Curriculum Course Description: AP Calculus BC is designed for the serious and motivated college bound student planning to major in math, science, or engineering. This course
More information1.10 Continuity Brian E. Veitch
1.10 Continuity Definition 1.5. A function is continuous at x = a if 1. f(a) exists 2. lim x a f(x) exists 3. lim x a f(x) = f(a) If any of these conditions fail, f is discontinuous. Note: From algebra
More informationMITOCW ocw f99-lec23_300k
MITOCW ocw-18.06-f99-lec23_300k -- and lift-off on differential equations. So, this section is about how to solve a system of first order, first derivative, constant coefficient linear equations. And if
More informationLearning Objectives for Math 166
Learning Objectives for Math 166 Chapter 6 Applications of Definite Integrals Section 6.1: Volumes Using Cross-Sections Draw and label both 2-dimensional perspectives and 3-dimensional sketches of the
More information2 = = 0 Thus, the number which is largest in magnitude is equal to the number which is smallest in magnitude.
Limits at Infinity Two additional topics of interest with its are its as x ± and its where f(x) ±. Before we can properly discuss the notion of infinite its, we will need to begin with a discussion on
More informationMath 162 Review of Series
Math 62 Review of Series. Explain what is meant by f(x) dx. What analogy (analogies) exists between such an improper integral and an infinite series a n? An improper integral with infinite interval of
More informationThe Growth of Functions. A Practical Introduction with as Little Theory as possible
The Growth of Functions A Practical Introduction with as Little Theory as possible Complexity of Algorithms (1) Before we talk about the growth of functions and the concept of order, let s discuss why
More informationAP Calculus Testbank (Chapter 9) (Mr. Surowski)
AP Calculus Testbank (Chapter 9) (Mr. Surowski) Part I. Multiple-Choice Questions n 1 1. The series will converge, provided that n 1+p + n + 1 (A) p > 1 (B) p > 2 (C) p >.5 (D) p 0 2. The series
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES In section 11.9, we were able to find power series representations for a certain restricted class of functions. INFINITE SEQUENCES AND SERIES
More information2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large. lim
Limits at Infinity and Horizontal Asymptotes As we prepare to practice graphing functions, we should consider one last piece of information about a function that will be helpful in drawing its graph the
More informationCalculus II Lecture Notes
Calculus II Lecture Notes David M. McClendon Department of Mathematics Ferris State University 206 edition Contents Contents 2 Review of Calculus I 5. Limits..................................... 7.2 Derivatives...................................3
More informationMATH 230 CALCULUS II OVERVIEW
MATH 230 CALCULUS II OVERVIEW This overview is designed to give you a brief look into some of the major topics covered in Calculus II. This short introduction is just a glimpse, and by no means the whole
More informationThe Not-Formula Book for C2 Everything you need to know for Core 2 that won t be in the formula book Examination Board: AQA
Not The Not-Formula Book for C Everything you need to know for Core that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More information1.4 Techniques of Integration
.4 Techniques of Integration Recall the following strategy for evaluating definite integrals, which arose from the Fundamental Theorem of Calculus (see Section.3). To calculate b a f(x) dx. Find a function
More informationA REVIEW OF RESIDUES AND INTEGRATION A PROCEDURAL APPROACH
A REVIEW OF RESIDUES AND INTEGRATION A PROEDURAL APPROAH ANDREW ARHIBALD 1. Introduction When working with complex functions, it is best to understand exactly how they work. Of course, complex functions
More informationLimits and Continuity
Limits and Continuity Philippe B. Laval Kennesaw State University January 2, 2005 Contents Abstract Notes and practice problems on its and continuity. Limits 2. Introduction... 2.2 Theory:... 2.2. GraphicalMethod...
More informationRolle s Theorem. The theorem states that if f (a) = f (b), then there is at least one number c between a and b at which f ' (c) = 0.
Rolle s Theorem Rolle's Theorem guarantees that there will be at least one extreme value in the interior of a closed interval, given that certain conditions are satisfied. As with most of the theorems
More information10.7 Trigonometric Equations and Inequalities
0.7 Trigonometric Equations and Inequalities 79 0.7 Trigonometric Equations and Inequalities In Sections 0., 0. and most recently 0., we solved some basic equations involving the trigonometric functions.
More informationAlgebra Exam. Solutions and Grading Guide
Algebra Exam Solutions and Grading Guide You should use this grading guide to carefully grade your own exam, trying to be as objective as possible about what score the TAs would give your responses. Full
More informationInstructor (Brad Osgood)
TheFourierTransformAndItsApplications-Lecture26 Instructor (Brad Osgood): Relax, but no, no, no, the TV is on. It's time to hit the road. Time to rock and roll. We're going to now turn to our last topic
More informationContinuity and One-Sided Limits
Continuity and One-Sided Limits 1. Welcome to continuity and one-sided limits. My name is Tuesday Johnson and I m a lecturer at the University of Texas El Paso. 2. With each lecture I present, I will start
More informationModule 9 : Infinite Series, Tests of Convergence, Absolute and Conditional Convergence, Taylor and Maclaurin Series
Module 9 : Infinite Series, Tests of Convergence, Absolute and Conditional Convergence, Taylor and Maclaurin Series Lecture 27 : Series of functions [Section 271] Objectives In this section you will learn
More information3.3 Limits and Infinity
Calculus Maimus. Limits Infinity Infinity is not a concrete number, but an abstract idea. It s not a destination, but a really long, never-ending journey. It s one of those mind-warping ideas that is difficult
More information2. FUNCTIONS AND ALGEBRA
2. FUNCTIONS AND ALGEBRA You might think of this chapter as an icebreaker. Functions are the primary participants in the game of calculus, so before we play the game we ought to get to know a few functions.
More informationSolutions to Math 41 First Exam October 18, 2012
Solutions to Math 4 First Exam October 8, 202. (2 points) Find each of the following its, with justification. If the it does not exist, explain why. If there is an infinite it, then explain whether it
More information