Chapter 9: Infinite Series Part 2

Transcription

1 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 of Functions by Power Series 9.10 Taylor and Maclaurin Series HW Sets Section 9.5: WebAssign Chapter 8 Set A Section 9.7: WebAssign Chapter 8 Set B Section 9.8: WebAssign Chapter 8 Set C Section 9.9: WebAssign Chapter 8 Set D Section 9.10: WebAssign Chapter 8 Set E 1

2 9.5 Alternating Series Continued Topics Use the Alternating Series Remainder to approximate the sum of an alternating series Using the Alternating Series Remainder to determine the number of terms required Warm Up! Determine the convergence or divergence of the following infinite series. a. b. #(% 1 #$% n #(% n 2 #+% 2

3 Example 1: Approximating the Sum of an Alternating Series Approximate the sum of the series by its first six terms. (Calculator Needed) a. b. #(% 1 #$% 1 n! 1 # 5 n! c. d. #(% 1 #$% 4 ln n + 1 #(% 1 #$% 2 n 3 e. #(% 1 #$% n 3 # 3

4 Example 2: Finding the Number of Terms Determine the number of terms required to approximate the sum of the series with an error of less than a. b. #(% 1 #$% n 5 #(% 1 #$% n 3 c. d. #(% 1 #$% n 6 #(% 1 #$% 2n 3 1 e. #(% 1 #$% n 7 4

5 9.7 Taylor Polynomials and Approximations Topics Find polynomial approximations of elementary functions and compare them with the elementary functions. Find Taylor and Maclaurin polynomial approximations of elementary functions. Use the remainder of a Taylor polynomial Warm Up! Go to 5

6 Example 1: First-Degree Polynomial Approximation Find a first-degree polynomial function P % whose value and slope agree with the value and slope of f at x = c. Verify graphically. a. f x = e >, c = 0 b. f x = > 5, c = 4 6

7 Example 2: Finding a Maclaurin Polynomial Find the nth Maclaurin polynomial for the function. If indicated, approximate the function at the given value of x using the polynomial found. a. f x = e 5>, n = 4, f % 5 b. f x = e +>, n = 5 7

8 c. f x = sin x, n = 5 d. f x = cos x, n = 4, f % 7 8

9 Example 3: Finding a Taylor Polynomial Find the nth Taylor polynomial for the function. If indicated, approximate the function at the given value of x using the polynomial found. a. f x = 6, n = 3, c = 1 > b. f x = ln x, n = 4, c = 2, f 2.1 9

10 Example 4: Determining the Accuracy of an Approximation Use Taylor s Theorem to obtain an upper bound for the error of the approximation. Then calculate the exact value of the error. a. sin % G 3! b. cos H 6! +..3 I 5! 10

11 c. e %H + %G + %I + %J 6! 3! 5! 7! Example 5: Approximating a Value to a Desired Accuracy Determine the degree of the Maclaurin polynomial required for the error in the approximation of the function at the indicated value of x to be less than a. f x = sin x, approximate f 0.3 b. f x = cos x, approximate f

12 c. f x = e >, approximate f 0.6 d. f x = ln x + 1, approximate f

13 9.8 Power Series Topics Understand the definition of a power series Find the radius of convergence Determine the endpoint convergence of a power series Differentiate and integrate a power series Warm Up! Using the Ratio Test, determine whether the following series converge or diverge. a. n! 3 # b. # c. 1 # (0.5) 6#$% 2n + 1! 13

14 Power Series - Introduction In the previous section, we saw that we can approximate functions using Taylor Polynomials. For example, e > 1 + x + x6 2! + x3 3! (the third degree Maclaurin polynomial approximation for f(x) = e > ) Using a power series, such functions can be represented exactly. A power series is an infinite series. For example, e > = 1 + x + x6 2! + x3 3! + + x# n! + It can be shown that the series on the right (of the equation above) converges to the number e >. Power Series - Definition If x is a variable, then an infinite series of the form O a # x # = a. + a % x + a 6 x 6 + a 3 x a # x # + Is called a power series. More generally, an infinite series of the form O a # (x c) # = a. + a % (x c) + a 6 (x c) 6 + a 3 (x c) a # (x c) # + Is called a power series centered at c, where c is a constant. 14

15 Example 1: Power Series Expand the first four terms of the following power series and state the value c which the series is centered around. a. x # n! b. 1 # x + 1 # c. #(% 1 n x 1 # 15

16 Example 2: Exploring the Radius of Convergence For each series and each given value of x, state whether the series converges or diverges. a. n! x # x = 1 x = 0 x = 1 b. 3 x 2 # x = 3 2 x = 2 x = 3 c. 1 # x # 2n + 1! x = 2 x = 0 x = 2 16

17 Radius and Interval of Convergence - Introduction A power series in x can be viewed as a function of x f(x) = O a # (x c) # where the domain of f(x) is the set of all x for which the power series converges. Determination of the domain of a power series is the primary concern in this section. Note: Every power series converges at its center c because f(c) = O = = So, c always lies in the domain of f. Theorem 9.20 (below) states that the domain of a power series can take three basic forms: a single point, an interval centered at c, or the entire real number line, as shown in Figure

18 Example 3: Finding the Radius of Convergence Find the radius of convergence of the following series using the Ratio Test. a. n! x # b. 3 x 2 # c. 1 # x # 2n + 1! 18

19 Interval of Convergence (Endpoint Convergence) For a power series whose radius of convergence is a finite number R, we have to determine the convergence or divergence of the endpoints of the interval of convergence. Each endpoint must be tested separately for convergence or divergence. As a result, the interval of convergence of a power series can take any one of the six forms shown below Example 4: Finding the Interval of Convergence Find the interval of convergence for each of the following series. a. b. n! x # 1 # x 6#$% 2n + 1! c = R = c = R = c. 3 x 2 # c = R = 19

20 d. x # n e. 1 # x + 1 # 2 # f. x # n 6 20

21 Extra Practice 1. Find the radius of convergence for the following power series a. c. 3x # 2n! x 3# n! b. 1 # x # 5 # 2. Find the interval of convergence for the following power series. a. b. 1 #$% n + 1 x # 3x # 2n! 21

22 c. d. 1 # x # n + 1 n + 2 x 3 #$% n #$% e. f. 1 # n! x 5 # 3 # 1 # x 6# n! 22

23 Power Series - Properties In many ways, a function defined by a power series behaves like a polynomial function. Properties: A function defined by power series is continuous in its interval of convergence. The derivative and antiderivative can be determined by differentiating and integrating each term of the power series. Example 5: Finding the Derivative Find the derivative of f(x). f x = x # n! 23

24 Example 6: Intervals of Convergence for f(x), f T x, and Consider the function f x = #(% x # n f(x) dx Expand the first four terms of each of the following series: a. f x dx b. f(x) c. f (x) Find the radius of convergence for each series. Find the interval of convergence for each of the following: a. f x dx b. f(x) c. f (x) 24

25 9.9 Representation of Functions by Power Series Topics Find a Geometric Power Series that represents a function Construct a power series using series operations. Warm Up! True or False? a. b. c. d. 25

26 Example 1: Finding a Geometric Power Series Centered at 0 Find a Geometric Power Series for the function centered at 0 then find the interval of convergence. a. f x = 5 >$6 b. f x = 6 7+> Example 2: Finding a Geometric Power Series Centered at 1. Find the Geometric Power Series for the function centered at 1. a. f x = % > 26

27 Example 3: Adding Two Power Series Find a power series for the function, centered at c, and determine the interval of convergence. a. f x = 3>+% > H +%, c = 0 b. f x = % X+>, c = 1 27

28 c. f x = 5> > H $6>+3, c = 0 Example 4: Finding a Power Series by Integration Use the power series 1 x = 1 # x 1 # Find a power series for the function centered at c. a. f x = ln x, c = 1 28

29 Example 5: Finding a Power Series by Integration Use the power series x = 1 # x #, x < 1 to find a power series for the function, centered at 0, and determine the interval of convergence. a. h x = +6 > H +% b. f x = 6 >$% G = [H [> H % >$% 29

30 c. f x = ln x + 1 = % >$% dx Example 6: Finding a Power Series by Integration Use the power series x = 1 # x # to find a power series for the function, centered at 0, and determine the interval of convergence. a. g x = arctan x 30

31 9.10 Taylor and Maclaurin Series Topics Find a Taylor or Maclaurin series for a function. Use a basic list of Taylor Series to find other Taylor Series. Warm Up! Find the nth Maclaurin Polynomial for the function. a. f x = sec x, n = 2 31

32 Example 1: Writing a Taylor Series Use the definition of Taylor Series to find the Taylor Series, centered at c, for the function. a. f x = e 6>, c = 0 b. f x = cos x, c = a 5 32

33 c. f x = % >, c = 1 d. f x = ln x, c = 1 33

34 Example 2: Forming a Power Series Use the function f x = sin x to form the Maclaurin series f # 0 n! x # = f 0 + f T 0 x + ftt 0 2! and determine the interval of convergence. x 6 + fttt 0 3! x 3 + f 5 0 4! x 5 + Example 3: A Convergent Maclaurin Series Prove that the Maclaurin series for the function converges to the function for all x. a. f x = sin x b. f x = cos x 34

35 Example 4: Maclaurin Series for a Composite Function Find the Maclaurin Series for the function given. a. f x = sin x 6 35

36 Example 5: Deriving a Power Series from a Basic List Find the Maclaurin Series for the function given. Use the table of power series for elementary functions. a. f x = cos x 36

37 b. f x = e >H /6 c. f x = sin πx d. f x = ln 1 + x 3 37

38 e. f x = sin 6 x f. f x = cos 6 x 38