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Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that R,(x) → 0.] Find the associated radius of convergence R. cos(3x) f(x) = 7

Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that R,(x) → 0.] Find the associated radius of convergence R. cos(3x) f(x) = 7

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that R,(x) → 0.] Find the associated radius of convergence R. cos(3x) f(x) = 7 Step 1 The Maclaurin series formula is f(x) = f(0) + f'(0)x + 2! f"(0) x2 f''(0) ,3 1 f(4)(0) x4 + .... 3! 4! The function f(x) cos(3x) has the following derivatives. %3D 7 sin (3r) 3 sin(3r) f'(x) = 7 9 cos (3x) |-zcos(3e) f"(x) = 27 f''(x) = sin( 3r) 27 sin(3r) 81 f(4)(x) cos (3x) 81 - cos(3x) 7
Transcribed Image Text:Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that R,(x) → 0.] Find the associated radius of convergence R. cos(3x) f(x) = 7 Step 1 The Maclaurin series formula is f(x) = f(0) + f'(0)x + 2! f"(0) x2 f''(0) ,3 1 f(4)(0) x4 + .... 3! 4! The function f(x) cos(3x) has the following derivatives. %3D 7 sin (3r) 3 sin(3r) f'(x) = 7 9 cos (3x) |-zcos(3e) f"(x) = 27 f''(x) = sin( 3r) 27 sin(3r) 81 f(4)(x) cos (3x) 81 - cos(3x) 7
Step 2 Now, f(0) = 1/7 1/7 , f'(0) = |-3/7 X , f"(0) = -9/7 -9/7 f''(0) = 27/7 X and f(4)(0) = 81/7 V 81/7 Step 3 Therefore, the Maclaurin Series begins as follows 1/7 V 1/7 x + 1/14 -9/14 x2 + ol x3 + |(No Response) + 27/56 x4 + -... Step 4 Σ Thus, the general formula is n = 0 Submit Skip (you cannot come back)
Transcribed Image Text:Step 2 Now, f(0) = 1/7 1/7 , f'(0) = |-3/7 X , f"(0) = -9/7 -9/7 f''(0) = 27/7 X and f(4)(0) = 81/7 V 81/7 Step 3 Therefore, the Maclaurin Series begins as follows 1/7 V 1/7 x + 1/14 -9/14 x2 + ol x3 + |(No Response) + 27/56 x4 + -... Step 4 Σ Thus, the general formula is n = 0 Submit Skip (you cannot come back)
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