Find the Taylor series centered at c = -1. f(x) = e4x Identify the correct expansion. 42-4 -(x + 1)" n! -4 4" e M8 M8 M8 M8 n=0 Σ n=0 -(x + 1)" n! x" e-4 n! 4" Σ(x + 1)" n=0 Find the interval on which the expansion is valid. (Give your answer as an interval in the form (*,*). Use the symbol ∞o for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")", "[","]" depending on whether the interval is open or closed. Enter Ø if the interval is empty. Express numbers in exact form. Use symbolic notation and fractions where needed.) interval: n=0
Find the Taylor series centered at c = -1. f(x) = e4x Identify the correct expansion. 42-4 -(x + 1)" n! -4 4" e M8 M8 M8 M8 n=0 Σ n=0 -(x + 1)" n! x" e-4 n! 4" Σ(x + 1)" n=0 Find the interval on which the expansion is valid. (Give your answer as an interval in the form (*,*). Use the symbol ∞o for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")", "[","]" depending on whether the interval is open or closed. Enter Ø if the interval is empty. Express numbers in exact form. Use symbolic notation and fractions where needed.) interval: n=0
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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![### Finding the Taylor Series and Interval of Convergence
#### Problem Statement
Find the Taylor series centered at \( c = -1 \) for the function
\[ f(x) = e^{4x} \]
Identify the correct expansion from the choices given.
#### Options for the Taylor Series Expansion
1. \[
\sum_{n=0}^{\infty} \frac{4^n - 4}{n!} (x + 1)^n
\]
2. \[
\sum_{n=0}^{\infty} \frac{4^n e^{-4}}{n!} (x + 1)^n
\]
3. \[
\sum_{n=0}^{\infty} \frac{x^n e^{-4}}{n!}
\]
4. \[
\sum_{n=0}^{\infty} \frac{4^n}{n!} (x + 1)^n
\]
#### Determine the Valid Interval of Expansion
Use the correct form of interval notation to express where the Taylor series expansion is valid. Options for interval notation include:
- Open intervals: \((a, b)\)
- Closed intervals: \([a, b]\)
- Mixed intervals: \((a, b]\) or \([a, b)\)
- Unbounded intervals using \(\infty\) for infinity
- Disjoint intervals combined with \(\cup\)
- Indicate if the interval is empty with \(\emptyset\)
Express numbers exactly and use symbolic notation and fractions where needed.
### Interval Input
\[ \text{interval}: \underline{\hspace{80%}} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1ca2b2e2-412c-4f12-8bdf-9b458afbcb62%2Fd8127ae7-501a-41f0-a1e6-adc0fd0d75ab%2Fj2khx67_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Taylor Series and Interval of Convergence
#### Problem Statement
Find the Taylor series centered at \( c = -1 \) for the function
\[ f(x) = e^{4x} \]
Identify the correct expansion from the choices given.
#### Options for the Taylor Series Expansion
1. \[
\sum_{n=0}^{\infty} \frac{4^n - 4}{n!} (x + 1)^n
\]
2. \[
\sum_{n=0}^{\infty} \frac{4^n e^{-4}}{n!} (x + 1)^n
\]
3. \[
\sum_{n=0}^{\infty} \frac{x^n e^{-4}}{n!}
\]
4. \[
\sum_{n=0}^{\infty} \frac{4^n}{n!} (x + 1)^n
\]
#### Determine the Valid Interval of Expansion
Use the correct form of interval notation to express where the Taylor series expansion is valid. Options for interval notation include:
- Open intervals: \((a, b)\)
- Closed intervals: \([a, b]\)
- Mixed intervals: \((a, b]\) or \([a, b)\)
- Unbounded intervals using \(\infty\) for infinity
- Disjoint intervals combined with \(\cup\)
- Indicate if the interval is empty with \(\emptyset\)
Express numbers exactly and use symbolic notation and fractions where needed.
### Interval Input
\[ \text{interval}: \underline{\hspace{80%}} \]
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