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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![Title: Finding the Maclaurin Series of \(\frac{\sin(x)}{x^8}\)
---
**Problem Statement:**
Find the Maclaurin series of \(\frac{\sin(x)}{x^8}\).
**Expression Format:**
\[
\sum_{n=0}^{\infty} \boxed{\phantom{element}}
\]
This problem requires finding the Maclaurin series (a type of Taylor series centered at 0) for the function given by \(\frac{\sin(x)}{x^8}\).
**Approach:**
1. Recall the Maclaurin series for \(\sin(x)\), which is given by:
\[
\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}
\]
2. Substitute this series expression into \(\frac{\sin(x)}{x^8}\).
3. Simplify the series by dividing each term by \(x^8\).
4. The resulting series will provide the desired Maclaurin series for the given function.
This process involves basic operations such as series manipulation and algebraic simplification.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62b092f3-63db-4d93-982f-67d0473d8e68%2F236903d7-71cd-485d-a061-632fcb86b8ec%2Fnhqh5fw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Title: Finding the Maclaurin Series of \(\frac{\sin(x)}{x^8}\)
---
**Problem Statement:**
Find the Maclaurin series of \(\frac{\sin(x)}{x^8}\).
**Expression Format:**
\[
\sum_{n=0}^{\infty} \boxed{\phantom{element}}
\]
This problem requires finding the Maclaurin series (a type of Taylor series centered at 0) for the function given by \(\frac{\sin(x)}{x^8}\).
**Approach:**
1. Recall the Maclaurin series for \(\sin(x)\), which is given by:
\[
\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}
\]
2. Substitute this series expression into \(\frac{\sin(x)}{x^8}\).
3. Simplify the series by dividing each term by \(x^8\).
4. The resulting series will provide the desired Maclaurin series for the given function.
This process involves basic operations such as series manipulation and algebraic simplification.
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