Differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f. 00 1 If f(x) = 1 5n %3D 25 n=0 00 f'(x) = n=1
Differentiate the given series expansion of f term-by-term to obtain the corresponding series expansion for the derivative of f. 00 1 If f(x) = 1 5n %3D 25 n=0 00 f'(x) = n=1
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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![**Topic: Differentiating a Power Series**
**Problem Statement:**
Differentiate the given series expansion of \( f \) term-by-term to obtain the corresponding series expansion for the derivative of \( f \).
**Given Function:**
\[
f(x) = \frac{1}{1 - x^5} = \sum_{n=0}^{\infty} x^{5n}
\]
**Objective:**
Find the derivative of \( f \), denoted as \( f'(x) \), using term-by-term differentiation of the series.
**Derivative Series Expansion:**
\[
f'(x) = \sum_{n=1}^{\infty} \boxed{}
\]
**Explanation of Series Expansion:**
The given series represents a geometric series where each term is of the form \( x^{5n} \). To find the derivative, differentiate each term separately with respect to \( x \). Adjust the indices and powers appropriately to ensure the series representation of the derivative is correct.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc6eea0f-cfa5-4eda-a078-cdf97fe23765%2F25ff6baa-d4d7-4708-ab73-a536836c6052%2Fjoyai0c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Differentiating a Power Series**
**Problem Statement:**
Differentiate the given series expansion of \( f \) term-by-term to obtain the corresponding series expansion for the derivative of \( f \).
**Given Function:**
\[
f(x) = \frac{1}{1 - x^5} = \sum_{n=0}^{\infty} x^{5n}
\]
**Objective:**
Find the derivative of \( f \), denoted as \( f'(x) \), using term-by-term differentiation of the series.
**Derivative Series Expansion:**
\[
f'(x) = \sum_{n=1}^{\infty} \boxed{}
\]
**Explanation of Series Expansion:**
The given series represents a geometric series where each term is of the form \( x^{5n} \). To find the derivative, differentiate each term separately with respect to \( x \). Adjust the indices and powers appropriately to ensure the series representation of the derivative is correct.
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