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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Question
![**Find the radius of convergence, \( R \), of the Maclaurin series of the function:**
\[ \ln(1 + x) \]
\[ R = \ \_\_\_\_\_\_\_\_ \]
---
This prompt is asking for the calculation of the radius of convergence of the Maclaurin series for the natural logarithm function, \( \ln(1 + x) \).
In general, the function \( \ln(1 + x) \) has a Maclaurin series which is given by:
\[ \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \]
To find the radius of convergence, \( R \), one can use the formula for the radius of convergence of a power series:
\[ R = \frac{1}{\limsup_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|} \]
For this series, \( R = 1 \), indicating that the series converges for \( |x| < 1 \).
This topic is crucial for understanding where a series representation of a function is valid, which is an important concept in calculus and analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fde5d5abe-d173-4c84-a521-ce5326c05407%2F20f4c361-66e9-4cd1-9072-ac8e037d66b1%2Fn022clb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Find the radius of convergence, \( R \), of the Maclaurin series of the function:**
\[ \ln(1 + x) \]
\[ R = \ \_\_\_\_\_\_\_\_ \]
---
This prompt is asking for the calculation of the radius of convergence of the Maclaurin series for the natural logarithm function, \( \ln(1 + x) \).
In general, the function \( \ln(1 + x) \) has a Maclaurin series which is given by:
\[ \ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \]
To find the radius of convergence, \( R \), one can use the formula for the radius of convergence of a power series:
\[ R = \frac{1}{\limsup_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|} \]
For this series, \( R = 1 \), indicating that the series converges for \( |x| < 1 \).
This topic is crucial for understanding where a series representation of a function is valid, which is an important concept in calculus and analysis.
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