Find the Taylor series centered at x = 0 for 2. 3. 1. (-1)¹ 42n 4. 5. 80 81 82 83 8 6. n = 0 n=0 ) n = 0 n=1 n=1 f(x) n = 0 n! (-1) 42n (2n)! (−1)n 4²n (2n)! 4n (2n)! 4n (2n)! An Pen = = cos(4). (-1)¹ 42n n! 22 xn Th p,2n
Find the Taylor series centered at x = 0 for 2. 3. 1. (-1)¹ 42n 4. 5. 80 81 82 83 8 6. n = 0 n=0 ) n = 0 n=1 n=1 f(x) n = 0 n! (-1) 42n (2n)! (−1)n 4²n (2n)! 4n (2n)! 4n (2n)! An Pen = = cos(4). (-1)¹ 42n n! 22 xn Th p,2n
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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![## Taylor Series Problem
### Objective:
Find the Taylor series centered at \( x = 0 \) for the function:
\[ f(x) = \cos(4x) \]
### Options:
**1.**
\[ \sum_{n=0}^{\infty} \frac{(-1)^n 4^{2n}}{n!} x^n \]
**2.**
\[ \sum_{n=0}^{\infty} \frac{(-1)^n 4^{2n}}{(2n)!} x^n \]
**3.**
\[ \sum_{n=0}^{\infty} \frac{(-1)^n 4^{2n}}{(2n)!} x^{2n} \]
**4.**
\[ \sum_{n=1}^{\infty} \frac{4^n}{(2n)!} x^n \]
**5.**
\[ \sum_{n=1}^{\infty} \frac{4^n}{(2n)!} x^{2n} \]
**6.**
\[ \sum_{n=0}^{\infty} \frac{(-1)^n 4^{2n}}{n!} x^{2n} \]
### Explanation:
Each option represents a potential Taylor series expansion for the function \(\cos(4x)\). You will need to determine which series corresponds to the actual expansion for \(\cos(4x)\) based on the properties of cosine and the standard Taylor series expansion formula.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a433dd5-6857-42c0-ae40-09bd40f154d3%2Fadd7af65-2ef2-4453-9545-1f2f0ebc1b8a%2F61aj1yhb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Taylor Series Problem
### Objective:
Find the Taylor series centered at \( x = 0 \) for the function:
\[ f(x) = \cos(4x) \]
### Options:
**1.**
\[ \sum_{n=0}^{\infty} \frac{(-1)^n 4^{2n}}{n!} x^n \]
**2.**
\[ \sum_{n=0}^{\infty} \frac{(-1)^n 4^{2n}}{(2n)!} x^n \]
**3.**
\[ \sum_{n=0}^{\infty} \frac{(-1)^n 4^{2n}}{(2n)!} x^{2n} \]
**4.**
\[ \sum_{n=1}^{\infty} \frac{4^n}{(2n)!} x^n \]
**5.**
\[ \sum_{n=1}^{\infty} \frac{4^n}{(2n)!} x^{2n} \]
**6.**
\[ \sum_{n=0}^{\infty} \frac{(-1)^n 4^{2n}}{n!} x^{2n} \]
### Explanation:
Each option represents a potential Taylor series expansion for the function \(\cos(4x)\). You will need to determine which series corresponds to the actual expansion for \(\cos(4x)\) based on the properties of cosine and the standard Taylor series expansion formula.
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