suppose Cn is a segn of real numbers such that lim Ich exists and is non-zero. If the ROC h→0 of the power series & Chah is equal to then ROC of the power serlis Σ n² ch is = ? chach n=1 n=1
suppose Cn is a segn of real numbers such that lim Ich exists and is non-zero. If the ROC h→0 of the power series & Chah is equal to then ROC of the power serlis Σ n² ch is = ? chach n=1 n=1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Title:** Problem Solving in Sequence and Series
**Content:**
Suppose \( C_n \) is a sequence of real numbers such that
\[ \lim_{n \to \infty} |C_n|^{1/n} \]
exists and is non-zero. If the Radius of Convergence (ROC) of the power series
\[ \sum_{n=1}^{\infty} C_n x^n \]
is equal to \( r \), then what is the ROC of the power series
\[ \sum_{n=1}^{\infty} n^2 C_n x^n \]?
**Explanation:**
- **Sequence \( C_n \)**: Given a sequence of real numbers \( C_n \).
- **Limit Condition**: The limit \( \lim_{n \to \infty} |C_n|^{1/n} \) exists and is non-zero.
- **Power Series 1**: \( \sum_{n=1}^{\infty} C_n x^n \) has a radius of convergence \( r \).
- **Power Series 2**: We are asked to determine the radius of convergence for the series \( \sum_{n=1}^{\infty} n^2 C_n x^n \).
In a power series, the radius of convergence (ROC) is determined by analyzing the limit of the nth term as \( n \) approaches infinity. For the series \( \sum_{n=1}^{\infty} C_n x^n \), the ROC is given by:
\[ r = \frac{1}{\limsup_{n \to \infty} |C_n|^{1/n}}. \]
Considering the augmented series \( \sum_{n=1}^{\infty} n^2 C_n x^n \), the presence of \( n^2 \) changes the ROC:
\[ \frac{1}{\limsup_{n \to \infty} |n^2 C_n|^{1/n}}. \]
Analyzing further, we have \( |n^2 C_n|^{1/n} = (n^2)^{1/n} |C_n|^{1/n} \), where \( (n^2)^{1/n} \rightarrow 1 \) as \( n \rightarrow \infty \). Thus, the ROC remains affected primarily by \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F588b9222-edf2-44c3-a5bc-71e88a46ec33%2F46d403b0-9737-4c78-8cd4-b47570c5e117%2Fann06da_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title:** Problem Solving in Sequence and Series
**Content:**
Suppose \( C_n \) is a sequence of real numbers such that
\[ \lim_{n \to \infty} |C_n|^{1/n} \]
exists and is non-zero. If the Radius of Convergence (ROC) of the power series
\[ \sum_{n=1}^{\infty} C_n x^n \]
is equal to \( r \), then what is the ROC of the power series
\[ \sum_{n=1}^{\infty} n^2 C_n x^n \]?
**Explanation:**
- **Sequence \( C_n \)**: Given a sequence of real numbers \( C_n \).
- **Limit Condition**: The limit \( \lim_{n \to \infty} |C_n|^{1/n} \) exists and is non-zero.
- **Power Series 1**: \( \sum_{n=1}^{\infty} C_n x^n \) has a radius of convergence \( r \).
- **Power Series 2**: We are asked to determine the radius of convergence for the series \( \sum_{n=1}^{\infty} n^2 C_n x^n \).
In a power series, the radius of convergence (ROC) is determined by analyzing the limit of the nth term as \( n \) approaches infinity. For the series \( \sum_{n=1}^{\infty} C_n x^n \), the ROC is given by:
\[ r = \frac{1}{\limsup_{n \to \infty} |C_n|^{1/n}}. \]
Considering the augmented series \( \sum_{n=1}^{\infty} n^2 C_n x^n \), the presence of \( n^2 \) changes the ROC:
\[ \frac{1}{\limsup_{n \to \infty} |n^2 C_n|^{1/n}}. \]
Analyzing further, we have \( |n^2 C_n|^{1/n} = (n^2)^{1/n} |C_n|^{1/n} \), where \( (n^2)^{1/n} \rightarrow 1 \) as \( n \rightarrow \infty \). Thus, the ROC remains affected primarily by \(
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
