Taylor and Maclaurin Series
|
|
- Cory York
- 5 years ago
- Views:
Transcription
1 Taylor and Maclaurin Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018
2 Background We have seen that some power series converge. When they do, we can think of them as functions, say f (x). The derivatives and antiderivatives of f (x) are power series too. Suppose we start with a function, f (x). Is there a power series representation for f (x)? Which functions have power series representations? How do we find the power series representation?
3 Differentiating a Power Series f (x) = a k (x c) k = a 0 + a 1 (x c) + f (c) = a 0 f (x) = k a k (x c) k 1 = a 1 + 2a 2 (x c) + k=1 f (c) = a 1 f (x) = k(k 1)a k (x c) k 2 = 2a 2 + 6a 3 (x c) + k=2 f (c) = 2a 2.
4 Differentiating a Power Series f (x) = a k (x c) k = a 0 + a 1 (x c) + f (c) = a 0 f (x) = k a k (x c) k 1 = a 1 + 2a 2 (x c) + k=1 f (c) = a 1 f (x) = k(k 1)a k (x c) k 2 = 2a 2 + 6a 3 (x c) + k=2 f (c) = 2a 2. Observation: in general f (k) (c) = a k a k = f (k) (c).
5 Summary If a k (x c) k converges with radius of convergence r > 0 to a function f (x), then f (x) = a k (x c) k = f (k) (c) (x c) k.
6 Big Question Question: suppose f (x) is infinitely differentiable, i.e., f (k) (x) exists for all k = 1, 2,..., then does f (k) (c) (x c) k converge? is its radius of convergence positive? does it converge to f (x)?
7 Big Question Question: suppose f (x) is infinitely differentiable, i.e., f (k) (x) exists for all k = 1, 2,..., then does f (k) (c) (x c) k converge? is its radius of convergence positive? does it converge to f (x)? If the answers are all yes, we say Taylor series expansion of f (x) about x = c. f (k) (c) (x c) k is the
8 Examples (1 of 3) Verify that e x. x k is the Taylor series expansion about c = 0 of
9 Examples (1 of 3) Verify that e x. x k is the Taylor series expansion about c = 0 of Let f (x) = e x, then f (k) (x) = e x for all k = 0, 1,.... If c = 0 then f (k) (c) = e c = e 0 = 1 for all k = 0, 1,.... Consequently f (k) (c) = 1 for all k = 0, 1,.... According to the Ratio Test, the power series f (k) (c) x k = x k converges absolutely for < x <, therefore e x = x k for all x.
10 Examples (2 of 3) Find Taylor series expansions for the following functions. 1. e 3x 2. e x 3. 2xe 3x
11 Examples (2 of 3) Find Taylor series expansions for the following functions. 1. e 3x (3x) k 3 k x k = = 2. e x 3. 2xe 3x
12 Examples (2 of 3) Find Taylor series expansions for the following functions. 1. e 3x (3x) k 3 k x k = = 2. e x = 3. 2xe 3x ( x) k = ( 1) k x k
13 Examples (2 of 3) Find Taylor series expansions for the following functions. 1. e 3x (3x) k 3 k x k = = 2. e x = ( x) k = 3. 2xe 3x = 2x ( 1) k x k ( 3x) k = 2( 3) k x k+1
14 Examples (3 of 3) Find an infinite series of positive terms whose sum is e.
15 Examples (3 of 3) Find an infinite series of positive terms whose sum is e. Since e x x k = for < x < then e 1 = 1 k = 1.
16 Terminology Definition A Taylor series for which c = 0, i.e., Maclaurin series. f (k) (0) x k is called a
17 Terminology Definition A Taylor series for which c = 0, i.e., Maclaurin series. f (k) (0) x k is called a Definition The n th partial sum of a Taylor series is polynomial of degree n P n (x) = n f (k) (c) (x c) k = f (c) + f (c)(x c) + + f (n) (c) (x c) n n! called the Taylor polynomial of degree n for f expanded about x = c.
18 Example Find the Taylor polynomials centered at c = 0 of degree 1, 3, and 5 for f (x) = sin x.
19 Example Find the Taylor polynomials centered at c = 0 of degree 1, 3, and 5 for f (x) = sin x. P 1 (x) = x P 3 (x) = x x 3! P 5 (x) = x x 3! + x 5 5!
20 Illustration y 3 -π - 2 π 3 - π π 3 2 π 3 π x sin x P 1 (x) P 3 (x) P 5 (x) -2-3
21 Taylor s Theorem Theorem (Taylor s Theorem) Suppose that f has n + 1 derivatives on the interval (c r, c + r) for some r > 0. Then for x (c r, c + r), f (x) P n (x) and the error in using P n (x) to approximate f (x) is R n (x) = f (x) P n (x) = f (n+1) (z) (x c)n+1 (n + 1)! for some z between x and c.
22 Taylor s Theorem Theorem (Taylor s Theorem) Suppose that f has n + 1 derivatives on the interval (c r, c + r) for some r > 0. Then for x (c r, c + r), f (x) P n (x) and the error in using P n (x) to approximate f (x) is R n (x) = f (x) P n (x) = f (n+1) (z) (x c)n+1 (n + 1)! for some z between x and c. Remarks: R n (x) is called the nth Taylor remainder. f (x) = P n (x) + R n (x).
23 Proof (1 of 4) (x t)n+1 n Define g(t) = f (x) R n (x) (x c) n+1 Verify that g(x) = 0 and that g(c) = 0. f (k) (t) (x t) k.
24 Proof (1 of 4) (x t)n+1 n Define g(t) = f (x) R n (x) (x c) n+1 Verify that g(x) = 0 and that g(c) = 0. f (k) (t) (x t) k. (x x)n+1 n g(x) = f (x) R n (x) (x c) n+1 (0) n+1 = f (x) R n (x) (x c) n+1 f (0) (x) 0! = f (x) f (x) = 0 f (k) (x) (x x) k n k=1 f (k) (x) (0) k
25 Proof (1 of 4) (x t)n+1 n Define g(t) = f (x) R n (x) (x c) n+1 Verify that g(x) = 0 and that g(c) = 0. f (k) (t) (x t) k. (x x)n+1 n g(x) = f (x) R n (x) (x c) n+1 (0) n+1 = f (x) R n (x) (x c) n+1 f (0) (x) 0! = f (x) f (x) = 0 (x c)n+1 n g(c) = f (x) R n (x) (x c) n+1 = f (x) R n (x) P n (x) = 0 f (k) (x) (x x) k n k=1 f (k) (c) (x c) k f (k) (x) (0) k
26 Proof (2 of 4) Since g(x) = 0 = g(c), g(t) is continuous on the closed interval from x to c, and g(t) is differentiable on the open interval from x to c, then by Rolle s Theorem there is a number z between x and c for which g (z) = 0. Differentiate g(t) with respect to t and set the derivative equal to zero.
27 Proof (3 of 4) (x t)n+1 n g(t) = f (x) R n (x) (x c) n+1 g ( 1)(n + 1)(x t)n (t) = R n (x) (x c) n+1 n f (k+1) (t) (x t) k (x t)n = (n + 1)R n (x) (x c) n+1 n f (k+1) (t) (x t) k + (x t)n = (n + 1)R n (x) (x c) n+1 n f (k+1) (t) f (k) (t) (x t) k n ( 1)k f (k) (t) (x t) k 1 k=1 n k=1 n 1 (x t) k + f (k) (t) (x t)k 1 (k 1)! f (k+1) (t) (x t) k
28 Proof (4 of 4) g (x t)n (t) = (n + 1)R n (x) (x c) n+1 n f (k+1) (t) n 1 (x t) k + f (k+1) (t) (x t) k g (x t)n (t) = (n + 1)R n (x) (x c) n+1 f (n+1) (t) (x t) n n! For some z between x and c, g (z) = 0, so (x z)n (n + 1)R n (x) (x c) n+1 f (n+1) (z) (x z) n = 0 n! 1 (n + 1)R n (x) (x c) n+1 = f (n+1) (z) n! R n (x) = f (n+1) (z) (n + 1)! (x c)n+1.
29 Big Answer Using Taylor s Theorem we can now answer the big questions posed at the beginning of this discussion.
30 Big Answer Using Taylor s Theorem we can now answer the big questions posed at the beginning of this discussion. Theorem If f (x) has derivatives of all orders in the interval (c r, c + r) for some r > 0 and if lim R n(x) = 0 for all x (c r, c + r), n then the Taylor series for f expanded about x = c converges to f (x), i.e., f (k) (c) f (x) = (x c) k.
31 Taylor s Inequality Theorem (Taylor s Inequality) If f (n+1) (x) M for x c r, then the remainder R n (x) of the Taylor series satisfies the inequality R n (x) M (n + 1)! x c n+1 for x c r.
32 Useful Result The following theorem will frequently be of use when trying to prove that lim n R n(x) = 0 for a potential Taylor series. Theorem If x is a real number then lim n x n n! = 0.
33 Verifying e x = x k We have seen that if f (x) = e x then a k = f (k) (0) = 1 and x k the power series converges (by the Ratio Test) for all x. An important detail is to show that it converges to e x. Let R n = f (n+1) (z) (n + 1)! x n+1 = between 0 and x. If x d then R n (x) = e z (n + 1)! x n+1 where z is e z (n + 1)! x n+1 lim R n(x) e d x n+1 lim n n (n + 1)! = 0. e d (n + 1)! x n+1
34 Example Find the Maclaurin series expansion for cos x.
35 Solution (1 of 2) Let f (x) = cos x, then f (0) = cos 0 = 1 f (0) = sin 0 = 0 f (0) = cos 0 = 1 f (0) = sin 0 = 0 f (4) (0) = cos 0 = 1 f (5) (0) = sin 0 = 0. f (n) (0) = 0 if n is odd, 1 if n/2 is even, 1 if n/2 is odd.
36 Solution (2 of 2) cos x = P n (x) + R n (x) = n ( 1) n (2n)! x 2n + = n d 2n+1 dx 2n+1 [cos x] x=z (2n + 1)! ( 1) n (2n)! x 2n + ± sin z (2n + 1)! x 2n+1 x 2n+1
37 Solution (2 of 2) Note that cos x = P n (x) + R n (x) = n ( 1) n (2n)! x 2n + = n d 2n+1 dx 2n+1 [cos x] x=z (2n + 1)! ( 1) n (2n)! x 2n + ± sin z (2n + 1)! x 2n+1 R n (x) = ± sin z (2n + 1)! x 2n+1 x 2n+1 (2n + 1)! lim R n(x) lim x 2n+1 n n (2n + 1)! = 0 x 2n+1
38 Error Estimates (1 of 4) 1. Find the Taylor series expansion for ln x about x = Estimate the error in using P 4 (x) to approximate ln Compare this with the actual error.
39 Error Estimates (2 of 4) Recall that d dx [ln x] = 1 x. 1 x = 1 1 (1 x) = (1 x) k = if 0 < x < 2. ( 1) k (x 1) k
40 Error Estimates (2 of 4) Recall that d dx [ln x] = 1 x. 1 x = 1 1 (1 x) = (1 x) k = if 0 < x < 2. Therefore ( 1) k (x 1) k ln x = ( 1) k k + 1 (x 1)k+1 = ( 1) k 1 (x 1) k k k=1
41 Error Estimates (2 of 4) Recall that d dx [ln x] = 1 x. 1 x = 1 1 (1 x) = (1 x) k = if 0 < x < 2. Therefore ( 1) k (x 1) k ln x = P 4 (x) = ( 1) k k + 1 (x 1)k+1 = 4 ( 1) k 1 (x 1) k k k=1 ( 1) k 1 (x 1) k k k=1 = (x 1) 1 2 (x 1) (x 1)3 1 (x 1)4 4 ln 1.2 P 4 (1.2)
42 Error Estimates (3 of 4) ( 1) k 1 Since (x 1) k is an alternating series then k k=1 ln 1.2 P 4 (1.2) 1 (1.2 1)
43 Error Estimates (3 of 4) ( 1) k 1 Since (x 1) k is an alternating series then k k=1 The actual error is ln 1.2 P 4 (1.2) 1 (1.2 1) ln 1.2 P 4 (1.2)
44 Error Estimates (4 of 4) Find the smallest value of n so that the maximum theoretical error in using P n (x) to approximate e x on the interval [ ln 10, ln 10] is less than 10 6.
45 Error Estimates (4 of 4) Find the smallest value of n so that the maximum theoretical error in using P n (x) to approximate e x on the interval [ ln 10, ln 10] is less than e z We can calculate R n (x) = (n + 1)! x n+1. R n (x) = e z (n + 1)! x n+1 eln 10 (n + 1)! (ln 10)n+1 when n 15 (found by trial and error). = 10 (n + 1)! (ln 10)n+1 < 10 6
46 Binomial Series Definition If n is any real number and x < 1, then (1+x) n = ( ) n x k = 1+n x+ k n(n 1) x 2 + 2! where the expressions ( ) n n(n 1)(n 2) (n k + 1) = k are called binomial coefficients. n(n 1)(n 2) x 3 + 3!
47 Example Find the Maclaurin series for f (x) = convergence. 1 1 x and its radius of
48 Example Find the Maclaurin series for f (x) = convergence. 1 1 x and its radius of Solution We can write f (x) = (1 x) 1/2 and use the Binomial Series formula with n = 1/2. ( ) 1/2 ( ) 1/2 f (x) = ( x) k = ( 1) k x k k k = 1 1 ( 1 ) ( 3 ) ( 1 ) ( 3 ) ( 5 ) 2 x x x 3 + 2! 3! = x + (1)(3) 2 2 2! x 2 + (1)(3)(5) 2 3 x 3 + 3!
49 Common Taylor Series Taylor series Interval of Convergence e x x k = (, ) ( 1) k x 2k+1 sin x = (, ) (2k + 1)! ( 1) k x 2k cos x = (, ) (2k)! ( 1) k+1 (x 1) k ln x = (0, 2] k k=1 tan 1 ( 1) k x 2k+1 x = ( 1, 1) 2k + 1 ( ) n (1 + x) n = x k ( 1, 1) k
50 Further Examples Find the Maclaurin series for the following functions. f (x) = (2 + x) 5 g(x) = ln 1 + x 1 x h(x) = x 3 sin x
51 Limits and Maclaurin Series Suppose f (x) = a k x k and g(x) = b k x k. Both Maclaurin series have a positive radius of convergence and f (0) = 0 = g(0). What does f (0) = 0 = g(0) imply about the coefficients of these two infinite series? f (x) Find lim x 0 g(x). Find lim x 0 cos(x 2 ) 1 ln(1 + x) x.
52 Function without Taylor Series Expansion Observation: Some functions have derivatives of all orders, but their Taylor remainders do not limit on 0, and thus there is no convergent Taylor series for these functions. Example Consider f (x) = { e 1/x 2 if x 0 0 if x = 0.
53 Function without Taylor Series Expansion Observation: Some functions have derivatives of all orders, but their Taylor remainders do not limit on 0, and thus there is no convergent Taylor series for these functions. Example Consider f (x) = { e 1/x 2 if x 0 0 if x = 0. We can show that f (k) (0) = 0 for all k = 0, 1, 2,.... f (k) (0) However f (x) 0 = x k.
54 Illustration f (x) = { e 1/x 2 if x 0 0 if x = 0. y x
55 Homework Read Section Exercises: WebAssign/D2L
Lecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain.
Lecture 32: Taylor Series and McLaurin series We saw last day that some functions are equal to a power series on part of their domain. For example f(x) = 1 1 x = 1 + x + x2 + x 3 + = ln(1 + x) = x x2 2
More informationSection Taylor and Maclaurin Series
Section.0 Taylor and Maclaurin Series Ruipeng Shen Feb 5 Taylor and Maclaurin Series Main Goal: How to find a power series representation for a smooth function us assume that a smooth function has a power
More informationReview of Power Series
Review of Power Series MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Introduction In addition to the techniques we have studied so far, we may use power
More informationJuly 21 Math 2254 sec 001 Summer 2015
July 21 Math 2254 sec 001 Summer 2015 Section 8.8: Power Series Theorem: Let a n (x c) n have positive radius of convergence R, and let the function f be defined by this power series f (x) = a n (x c)
More informationTAYLOR AND MACLAURIN SERIES
TAYLOR AND MACLAURIN SERIES. Introduction Last time, we were able to represent a certain restricted class of functions as power series. This leads us to the question: can we represent more general functions
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 informationCompletion Date: Monday February 11, 2008
MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #4 Completion Date: Monday February, 8 Department of Mathematical and Statistical Sciences University of Alberta Question. [Sec..9,
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 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 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 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 informationSection 10.7 Taylor series
Section 10.7 Taylor series 1. Common Maclaurin series 2. s and approximations with Taylor polynomials 3. Multiplication and division of power series Math 126 Enhanced 10.7 Taylor Series The University
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 informationTaylor Series. Math114. March 1, Department of Mathematics, University of Kentucky. Math114 Lecture 18 1/ 13
Taylor Series Math114 Department of Mathematics, University of Kentucky March 1, 2017 Math114 Lecture 18 1/ 13 Given a function, can we find a power series representation? Math114 Lecture 18 2/ 13 Given
More informationSection 8.7. Taylor and MacLaurin Series. (1) Definitions, (2) Common Maclaurin Series, (3) Taylor Polynomials, (4) Applications.
Section 8.7 Taylor and MacLaurin Series (1) Definitions, (2) Common Maclaurin Series, (3) Taylor Polynomials, (4) Applications. MATH 126 (Section 8.7) Taylor and MacLaurin Series The University of Kansas
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 informationMath 651 Introduction to Numerical Analysis I Fall SOLUTIONS: Homework Set 1
ath 651 Introduction to Numerical Analysis I Fall 2010 SOLUTIONS: Homework Set 1 1. Consider the polynomial f(x) = x 2 x 2. (a) Find P 1 (x), P 2 (x) and P 3 (x) for f(x) about x 0 = 0. What is the relation
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 informationMath 227 Sample Final Examination 1. Name (print) Name (sign) Bing ID number
Math 227 Sample Final Examination 1 Name (print) Name (sign) Bing ID number (Your instructor may check your ID during or after the test) No books, notes, or electronic devices (calculators, cell phones,
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 information10.1 Sequences. Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1.
10.1 Sequences Example: A sequence is a function f(n) whose domain is a subset of the integers. Notation: *Note: n = 0 vs. n = 1 Examples: EX1: Find a formula for the general term a n of the sequence,
More informationMath Review for Exam Answer each of the following questions as either True or False. Circle the correct answer.
Math 22 - Review for Exam 3. Answer each of the following questions as either True or False. Circle the correct answer. (a) True/False: If a n > 0 and a n 0, the series a n converges. Soln: False: Let
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 informationSection Example Determine the Maclaurin series of f (x) = e x and its the interval of convergence.
Example Determine the Maclaurin series of f (x) = e x and its the interval of convergence. Example Determine the Maclaurin series of f (x) = e x and its the interval of convergence. f n (0)x n Recall from
More informationMath 115 HW #5 Solutions
Math 5 HW #5 Solutions From 29 4 Find the power series representation for the function and determine the interval of convergence Answer: Using the geometric series formula, f(x) = 3 x 4 3 x 4 = 3(x 4 )
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 informationStudy # 1 11, 15, 19
Goals: 1. Recognize Taylor Series. 2. Recognize the Maclaurin Series. 3. Derive Taylor series and Maclaurin series representations for known functions. Study 11.10 # 1 11, 15, 19 f (n) (c)(x c) n f(c)+
More informationCALCULUS JIA-MING (FRANK) LIOU
CALCULUS JIA-MING (FRANK) LIOU Abstract. Contents. Power Series.. Polynomials and Formal Power Series.2. Radius of Convergence 2.3. Derivative and Antiderivative of Power Series 4.4. Power Series Expansion
More informationAbsolute Convergence and the Ratio Test
Absolute Convergence and the Ratio Test MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Bacground Remar: All previously covered tests for convergence/divergence apply only
More informationAP Calculus (BC) Chapter 9 Test No Calculator Section Name: Date: Period:
WORKSHEET: Series, Taylor Series AP Calculus (BC) Chapter 9 Test No Calculator Section Name: Date: Period: 1 Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The
More informationSection 9.7 and 9.10: Taylor Polynomials and Approximations/Taylor and Maclaurin Series
Section 9.7 and 9.10: Taylor Polynomials and Approximations/Taylor and Maclaurin Series Power Series for Functions We can create a Power Series (or polynomial series) that can approximate a function around
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 informationMAT137 Calculus! Lecture 45
official website http://uoft.me/mat137 MAT137 Calculus! Lecture 45 Today: Taylor Polynomials Taylor Series Next: Taylor Series Power Series Definition (Power Series) A power series is a series of the form
More informatione x = 1 + x + x2 2! + x3 If the function f(x) can be written as a power series on an interval I, then the power series is of the form
Taylor Series Given a function f(x), we would like to be able to find a power series that represents the function. For example, in the last section we noted that we can represent e x by the power series
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 informationAbsolute Convergence and the Ratio Test
Absolute Convergence and the Ratio Test MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Bacground Remar: All previously covered tests for convergence/divergence apply only
More informationProperties of a Taylor Polynomial
3.4.4: Still Better Approximations: Taylor Polynomials Properties of a Taylor Polynomial Constant: f (x) f (a) Linear: f (x) f (a) + f (a)(x a) Quadratic: f (x) f (a) + f (a)(x a) + 1 2 f (a)(x a) 2 3.4.4:
More informationContinuity. MATH 161 Calculus I. J. Robert Buchanan. Fall Department of Mathematics
Continuity MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Fall 2017 Intuitive Idea A process or an item can be described as continuous if it exists without interruption. The mathematical
More informationReview: Power series define functions. Functions define power series. Taylor series of a function. Taylor polynomials of a function.
Taylor Series (Sect. 10.8) Review: Power series define functions. Functions define power series. Taylor series of a function. Taylor polynomials of a function. Review: Power series define functions Remarks:
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 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 informationChapter 11. Taylor Series. Josef Leydold Mathematical Methods WS 2018/19 11 Taylor Series 1 / 27
Chapter 11 Taylor Series Josef Leydold Mathematical Methods WS 2018/19 11 Taylor Series 1 / 27 First-Order Approximation We want to approximate function f by some simple function. Best possible approximation
More informationHAND IN PART. Prof. Girardi Math 142 Spring Exam 3 PIN:
HAND IN PART Prof. Girardi Math 142 Spring 2014 04.17.2014 Exam 3 MARK BOX problem points possible your score 0A 9 0B 8 0C 10 0D 12 NAME: PIN: solution key Total for 0 39 Total for 1 10 61 % 100 INSTRUCTIONS
More informationMATH 1231 MATHEMATICS 1B Calculus Section 4.4: Taylor & Power series.
MATH 1231 MATHEMATICS 1B 2010. For use in Dr Chris Tisdell s lectures. Calculus Section 4.4: Taylor & Power series. 1. What is a Taylor series? 2. Convergence of Taylor series 3. Common Maclaurin series
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 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 information8.5 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 informationChapter 8: Taylor s theorem and L Hospital s rule
Chapter 8: Taylor s theorem and L Hospital s rule Theorem: [Inverse Mapping Theorem] Suppose that a < b and f : [a, b] R. Given that f (x) > 0 for all x (a, b) then f 1 is differentiable on (f(a), f(b))
More informationSOLVED PROBLEMS ON TAYLOR AND MACLAURIN SERIES
SOLVED PROBLEMS ON TAYLOR AND MACLAURIN SERIES TAYLOR AND MACLAURIN SERIES Taylor Series of a function f at x = a is ( f k )( a) ( x a) k k! It is a Power Series centered at a. Maclaurin Series of a function
More informationMAT137 Calculus! Lecture 48
official website http://uoft.me/mat137 MAT137 Calculus! Lecture 48 Today: Taylor Series Applications Next: Final Exams Important Taylor Series and their Radii of Convergence 1 1 x = e x = n=0 n=0 x n n!
More informationTaylor Series and Maclaurin Series
Taylor Series and Maclaurin Series Definition (Taylor Series) Suppose the function f is infinitely di erentiable at a. The Taylor series of f about a (or at a or centered at a) isthepowerseries f (n) (a)
More informationMath 181, Exam 2, Study Guide 2 Problem 1 Solution. 1 + dx. 1 + (cos x)2 dx. 1 + cos2 xdx. = π ( 1 + cos π 2
Math 8, Exam, Study Guide Problem Solution. Use the trapezoid rule with n to estimate the arc-length of the curve y sin x between x and x π. Solution: The arclength is: L b a π π + ( ) dy + (cos x) + cos
More informationPart 3.3 Differentiation Taylor Polynomials
Part 3.3 Differentiation 3..3.1 Taylor Polynomials Definition 3.3.1 Taylor 1715 and Maclaurin 1742) If a is a fixed number, and f is a function whose first n derivatives exist at a then the Taylor polynomial
More informationMATH 163 HOMEWORK Week 13, due Monday April 26 TOPICS. c n (x a) n then c n = f(n) (a) n!
MATH 63 HOMEWORK Week 3, due Monday April 6 TOPICS 4. Taylor series Reading:.0, pages 770-77 Taylor series. If a function f(x) has a power series representation f(x) = c n (x a) n then c n = f(n) (a) ()
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 informationSection 5.8. Taylor Series
Difference Equations to Differential Equations Section 5.8 Taylor Series In this section we will put together much of the work of Sections 5.-5.7 in the context of a discussion of Taylor series. We begin
More informationMath 1552: Integral Calculus Final Exam Study Guide, Spring 2018
Math 55: Integral Calculus Final Exam Study Guide, Spring 08 PART : Concept Review (Note: concepts may be tested on the exam in the form of true/false or short-answer questions.). Complete each statement
More informationf (x) = k=0 f (0) = k=0 k=0 a k k(0) k 1 = a 1 a 1 = f (0). a k k(k 1)x k 2, k=2 a k k(k 1)(0) k 2 = 2a 2 a 2 = f (0) 2 a k k(k 1)(k 2)x k 3, k=3
1 M 13-Lecture Contents: 1) Taylor Polynomials 2) Taylor Series Centered at x a 3) Applications of Taylor Polynomials Taylor Series The previous section served as motivation and gave some useful expansion.
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 informationInfinite Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Infinite Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Background Consider the repeating decimal form of 2/3. 2 3 = 0.666666 = 0.6 + 0.06 + 0.006 + 0.0006 + = 6(0.1)
More informationFinal exam (practice) UCLA: Math 31B, Spring 2017
Instructor: Noah White Date: Final exam (practice) UCLA: Math 3B, Spring 207 This exam has 8 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions in the
More informationPower Series. x n. Using the ratio test. n n + 1. x n+1 n 3. = lim x. lim n + 1. = 1 < x < 1. Then r = 1 and I = ( 1, 1) ( 1) n 1 x n.
.8 Power Series. n x n x n n Using the ratio test. lim x n+ n n + lim x n n + so r and I (, ). By the ratio test. n Then r and I (, ). n x < ( ) n x n < x < n lim x n+ n (n + ) x n lim xn n (n + ) x
More informationFundamental Theorem of Calculus
Fundamental Theorem of Calculus MATH 6 Calculus I J. Robert Buchanan Department of Mathematics Summer 208 Remarks The Fundamental Theorem of Calculus (FTC) will make the evaluation of definite integrals
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 informationConstructing Taylor Series
Constructing Taylor Series 8-8-200 The Taylor series for fx at x = c is fc + f cx c + f c 2! x c 2 + f c x c 3 + = 3! f n c x c n. By convention, f 0 = f. When c = 0, the series is called a Maclaurin series.
More informationAnalysis II: Basic knowledge of real analysis: Part V, Power Series, Differentiation, and Taylor Series
.... Analysis II: Basic knowledge of real analysis: Part V, Power Series, Differentiation, and Taylor Series Kenichi Maruno Department of Mathematics, The University of Texas - Pan American March 4, 20
More informationMath 1B, lecture 15: Taylor Series
Math B, lecture 5: Taylor Series Nathan Pflueger October 0 Introduction Taylor s theorem shows, in many cases, that the error associated with a Taylor approximation will eventually approach 0 as the degree
More informationCalculus I Sample Exam #01
Calculus I Sample Exam #01 1. Sketch the graph of the function and define the domain and range. 1 a) f( x) 3 b) g( x) x 1 x c) hx ( ) x x 1 5x6 d) jx ( ) x x x 3 6 . Evaluate the following. a) 5 sin 6
More information1 Review of di erential calculus
Review of di erential calculus This chapter presents the main elements of di erential calculus needed in probability theory. Often, students taking a course on probability theory have problems with concepts
More informationWorksheet 9. Topics: Taylor series; using Taylor polynomials for approximate computations. Polar coordinates.
ATH 57H Spring 0 Worksheet 9 Topics: Taylor series; using Taylor polynomials for approximate computations. Polar coordinates.. Let f(x) = +x. Find f (00) (0) - the 00th derivative of f at point x = 0.
More informationMath 221 Notes on Rolle s Theorem, The Mean Value Theorem, l Hôpital s rule, and the Taylor-Maclaurin formula. 1. Two theorems
Math 221 Notes on Rolle s Theorem, The Mean Value Theorem, l Hôpital s rule, and the Taylor-Maclaurin formula 1. Two theorems Rolle s Theorem. If a function y = f(x) is differentiable for a x b and if
More informationRepresentation of Functions as Power Series
Representation of Functions as Power Series Philippe B. Laval KSU Today Philippe B. Laval (KSU) Functions as Power Series Today / Introduction In this section and the next, we develop several techniques
More informationRepresentation of Functions as Power Series.
MATH 0 - A - Spring 009 Representation of Functions as Power Series. Our starting point in this section is the geometric series: x n = + x + x + x 3 + We know this series converges if and only if x
More informationIntroduction and Review of Power Series
Introduction and Review of Power Series Definition: A power series in powers of x a is an infinite series of the form c n (x a) n = c 0 + c 1 (x a) + c 2 (x a) 2 +...+c n (x a) n +... If a = 0, this is
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 informationMATH115. Infinite Series. Paolo Lorenzo Bautista. July 17, De La Salle University. PLBautista (DLSU) MATH115 July 17, / 43
MATH115 Infinite Series Paolo Lorenzo Bautista De La Salle University July 17, 2014 PLBautista (DLSU) MATH115 July 17, 2014 1 / 43 Infinite Series Definition If {u n } is a sequence and s n = u 1 + u 2
More informationYou can learn more about the services offered by the teaching center by visiting
MAC 232 Exam 3 Review Spring 209 This review, produced by the Broward Teaching Center, contains a collection of questions which are representative of the type you may encounter on the exam. Other resources
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 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 informationChain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics
3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)
More informationMATH 1372, SECTION 33, MIDTERM 3 REVIEW ANSWERS
MATH 1372, SECTION 33, MIDTERM 3 REVIEW ANSWERS 1. We have one theorem whose conclusion says an alternating series converges. We have another theorem whose conclusion says an alternating series diverges.
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 informationMath 113 Winter 2005 Key
Name Student Number Section Number Instructor Math Winter 005 Key Departmental Final Exam Instructions: The time limit is hours. Problem consists of short answer questions. Problems through are multiple
More informationMATH 1242 FINAL EXAM Spring,
MATH 242 FINAL EXAM Spring, 200 Part I (MULTIPLE CHOICE, NO CALCULATORS).. Find 2 4x3 dx. (a) 28 (b) 5 (c) 0 (d) 36 (e) 7 2. Find 2 cos t dt. (a) 2 sin t + C (b) 2 sin t + C (c) 2 cos t + C (d) 2 cos t
More informationChapter 9: Infinite Series Part 2
Name: Date: Period: AP Calc BC Mr. Mellina/Ms. Lombardi Chapter 9: Infinite Series Part 2 Topics: 9.5 Alternating Series Remainder 9.7 Taylor Polynomials and Approximations 9.8 Power Series 9.9 Representation
More information, applyingl Hospital s Rule again x 0 2 cos(x) xsinx
Lecture 3 We give a couple examples of using L Hospital s Rule: Example 3.. [ (a) Compute x 0 sin(x) x. To put this into a form for L Hospital s Rule we first put it over a common denominator [ x 0 sin(x)
More informationTAYLOR SERIES [SST 8.8]
TAYLOR SERIES [SST 8.8] TAYLOR SERIES: Every function f C (c R, c + R) has a unique Taylor series about x = c of the form: f (k) (c) f(x) = (x c) k = f(c) + f (c) (x c) + f (c) (x c) 2 + f (c) (x c) 3
More informationMean Value Theorem. MATH 161 Calculus I. J. Robert Buchanan. Summer Department of Mathematics
Mean Value Theorem MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Background: Corollary to the Intermediate Value Theorem Corollary Suppose f is continuous on the closed interval
More informationMean Value Theorem. MATH 161 Calculus I. J. Robert Buchanan. Summer Department of Mathematics
Mean Value Theorem MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Background: Corollary to the Intermediate Value Theorem Corollary Suppose f is continuous on the closed interval
More informationAdvanced Calculus Math 127B, Winter 2005 Solutions: Final. nx2 1 + n 2 x, g n(x) = n2 x
. Define f n, g n : [, ] R by f n (x) = Advanced Calculus Math 27B, Winter 25 Solutions: Final nx2 + n 2 x, g n(x) = n2 x 2 + n 2 x. 2 Show that the sequences (f n ), (g n ) converge pointwise on [, ],
More information1 Antiderivatives graphically and numerically
Math B - Calculus by Hughes-Hallett, et al. Chapter 6 - Constructing antiderivatives Prepared by Jason Gaddis Antiderivatives graphically and numerically Definition.. The antiderivative of a function f
More informationEngg. Math. I. Unit-I. Differential Calculus
Dr. Satish Shukla 1 of 50 Engg. Math. I Unit-I Differential Calculus Syllabus: Limits of functions, continuous functions, uniform continuity, monotone and inverse functions. Differentiable functions, Rolle
More informationSeries Solutions. 8.1 Taylor Polynomials
8 Series Solutions 8.1 Taylor Polynomials Polynomial functions, as we have seen, are well behaved. They are continuous everywhere, and have continuous derivatives of all orders everywhere. It also turns
More informationSection 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that
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 informationThe polar coordinates
The polar coordinates 1 2 3 4 Graphing in polar coordinates 5 6 7 8 Area and length in polar coordinates 9 10 11 Partial deravitive 12 13 14 15 16 17 18 19 20 Double Integral 21 22 23 24 25 26 27 Triple
More informationCalculus and Parametric Equations
Calculus and Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Given a pair a parametric equations x = f (t) y = g(t) for a t b we know how
More informationSeries Solutions of Differential Equations
Chapter 6 Series Solutions of Differential Equations In this chapter we consider methods for solving differential equations using power series. Sequences and infinite series are also involved in this treatment.
More informationThis practice exam is intended to help you prepare for the final exam for MTH 142 Calculus II.
MTH 142 Practice Exam Chapters 9-11 Calculus II With Analytic Geometry Fall 2011 - University of Rhode Island This practice exam is intended to help you prepare for the final exam for MTH 142 Calculus
More informationIntroduction Derivation General formula List of series Convergence Applications Test SERIES 4 INU0114/514 (MATHS 1)
MACLAURIN SERIES SERIES 4 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Maclaurin Series 1/ 21 Adrian Jannetta Recap: Binomial Series Recall that some functions can be rewritten as a power series
More information