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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![**Problem Statement:**
Find the Maclaurin series of \( 9^x \).
**Mathematical Representation:**
\[
\sum_{n=0}^{\infty} \, \_\_
\]
**Explanation:**
In the image, there is an incomplete mathematical expression for representing the Maclaurin series of \( 9^x \). The series expansion begins with the summation symbol \(\sum\) that shows the sum of an infinite series starting from \( n = 0 \) to \( \infty \). Typically, one would need to fill in the series terms in the space provided to express the function \( 9^x \) as an infinite series.
The Maclaurin series is a special case of the Taylor series, centered around zero. To calculate the series, you would generally derive the function multiple times, evaluate each derivative at zero, and substitute these values into the series formula:
\[
f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots
\]
For \( 9^x \), these derivatives will help determine the coefficients of each term in the series expansion.
This type of task typically provides practice in series expansion techniques commonly used in calculus and mathematical analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1dbff557-c987-45fc-8dd6-6084416c9157%2F00b30923-a453-46cc-bf66-65cfa720af6c%2F2c0iuw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the Maclaurin series of \( 9^x \).
**Mathematical Representation:**
\[
\sum_{n=0}^{\infty} \, \_\_
\]
**Explanation:**
In the image, there is an incomplete mathematical expression for representing the Maclaurin series of \( 9^x \). The series expansion begins with the summation symbol \(\sum\) that shows the sum of an infinite series starting from \( n = 0 \) to \( \infty \). Typically, one would need to fill in the series terms in the space provided to express the function \( 9^x \) as an infinite series.
The Maclaurin series is a special case of the Taylor series, centered around zero. To calculate the series, you would generally derive the function multiple times, evaluate each derivative at zero, and substitute these values into the series formula:
\[
f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots
\]
For \( 9^x \), these derivatives will help determine the coefficients of each term in the series expansion.
This type of task typically provides practice in series expansion techniques commonly used in calculus and mathematical analysis.
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