Section 10.9 Reading Assignment: Convergence of Taylor Series

Answer Only Exercise 1, 2, and 3 by using a screenshot provided Calculus Pearson textbook. Make sure you read these three questions very carefully and see on what it is asking for and what is really about. Please be very serious careful with this assignment of exercise #1, #2, and #3 or else I have to report the answer.

References: Thomas' Calculus: Early Transcendentals | Calculus | Calculus | Mathematics | Store | Pearson+

Section 10.9 Reading Assignment: Convergence of Taylor Series Instructions. Read through this assignment and complete the three exercises below by reading the appropriate passages of the textbook. This section begins with some discussion about the remainder of Taylor polynomials. The book's immediate concern is to ensure that the Taylor series is equal to the original function. We're not as concerned about that particular issue, but we will return to the idea of estimating the remainder. Our main focus is on the idea of using power series operations (first covered in Section 10.7) to obtain new Taylor series from more basic ones. Exercise 1. Read Example 4(a) (p. 644), skipping part (b). Explain the process for multiplying = x by the power series. Write out the power series for = x cos(x) in Sigma notation (ignore the addition by = x in this example). Exercise 2. Read the paragraph after Example 4 but before Example 5 (p. 644 - 645). Explain the process by which the textbook obtains the Taylor series for cos(2x). These power series operations are quite important since they are used to obtain many new Taylor series. These methods are easier than they may first seem, but historically many students have difficulties with this, so it's best to spend a bit of time getting familiar with these methods. In the next section, we'll also recall the idea of term-by-term integration, which is especially useful.Exercise 3. Read Example 5 (p. 645). Explain what is being found in this example and how it's done. Be sure to include the name of the theorem used in this example and why it applies. While there is a more general error estimation covered earlier in the form of The Remainder Estimation Theorem (p. 643). That one tends to be a bit harder to use due to the value of M. You can skip the proof of Taylor's theorem. It won't be important for us. For the record, the proof is repeated use of the mean value theorem from Calculus I.Chapter 10 Infinite Sequences and Series 644 Chapter 10 Infinite Sequences and Series EXAMPLE 3 Show that the Taylor series for cos x at x = 0 converges to cos x for every value of x. Solution We add the remainder term to the Taylor polynomial for cos x (Section 10.8, Example 3) to obtain Taylor's formula for cos x with n = 2k: COs x = 1 - 2! 4! (24)! 7+ Rx(x). Because the derivatives of the cosine have absolute value less than or equal to 1, the Remainder Estimation Theorem with M = 1 gives | Ry(x) | $ 1 . 1x/24+1 (2k + 1)!' For every value of x, Ry(x) -0 as k-co. Therefore, the series converges to cos x for every value of r. Thus, COS X = 2! 6! (5) COS * = 1 - 4! 21 Using Taylor Series Since every Taylor series is a power series, the operations of adding, subtracting, and mul- tiplying Taylor series are all valid on the intersection of their intervals of convergence. EXAMPLE 4 Using known series, find the first few terms of the Taylor series for the given function by using power series operations. (a) (2x + x cos x) (b) e cos xSolution 7-24 Taylor series (a) = (2x + x cos x) = x + X + + 4! (24)! for COS X = X + EX- + 3! 3 -4! 6 72 Multiply the first (b) e cosx = 2! + + series by each term 3! 2! + 4! of the second series. + + + + + 3! 4! 21 2! 212! 213! + ... 214! =1+x- 6 + .. By Theorem 20, we can use the Taylor series of the function f to find the Taylor series of f(e(x)) where u(x) is any continuous function. The Taylor series resulting from this substitution will converge for all x such that u(x) lies within the interval of convergence ofChapter 10 Infinite Sequences and Series 10.9 Convergence of Taylor Series 645 the Taylor series of f. For instance, we can find the Taylor series for cos 2.x by substituting 2x for x in the Taylor series for cos r: (-1)'(2r)2 cos 2x = = 1 - (20)2 (2r)4 (2r) (24)! 2! + . .. 4! 6! Eq. (5) with 2x for x = 1- 242 2414 2! + 4! 6! it . .. (24)! ' EXAMPLE 5 For what values of x can we replace sin x by x - (x]/3!) and obtain an error whose magnitude is no greater than 3 x 10-4? Solution Here we can take advantage of the fact that the Taylor series for sinx is an alternating series for every nonzero value of x. According to the Alternating Series Estima- tion Theorem (Section 10.6), the error in truncating sin x = x - 3! ! - 71 + ... after (x /3!) is no greater than 5! = 120' Therefore the error will be less than or equal to 3 X 10 4 if 120 0 as k ->co, whatever the value of x, so Ry+ (x) -0 and the Maclaurin series for sin x converges to sin x for every x. Thus. (-1)43+1 sin x = : + sin x = IME 7! (24 + 1)! =X 71 (4)