Use the root test to find the radius of convergence for 9n² – 1 00 2". n² + 8
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Problem Statement:**
Use the root test to find the radius of convergence for
\[
\sum_{n=1}^{\infty} \left(\frac{9n^2 - 1}{n^2 + 8}\right)^n x^n.
\]
**Explanation:**
This problem involves using the root test to determine the radius of convergence for the given power series. The power series in question has the general term:
\[
a_n = \left(\frac{9n^2 - 1}{n^2 + 8}\right)^n x^n.
\]
The root test for convergence involves analyzing the limit:
\[
\lim_{n \to \infty} \sqrt[n]{|a_n|}.
\]
By calculating this limit, you can determine the value of \(x\) for which the series is convergent. The radius of convergence \(R\) can then be found using the relation that the series converges if:
\[
\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1 \implies |x| < R.
\]
The challenge is to evaluate this limit and solve for \(R\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcfea0b1d-fbd9-4aaf-b240-b46cb35251fe%2F144aa75e-b2b7-4cea-b888-ae00449dd5e3%2Fwtbzi9o_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use the root test to find the radius of convergence for
\[
\sum_{n=1}^{\infty} \left(\frac{9n^2 - 1}{n^2 + 8}\right)^n x^n.
\]
**Explanation:**
This problem involves using the root test to determine the radius of convergence for the given power series. The power series in question has the general term:
\[
a_n = \left(\frac{9n^2 - 1}{n^2 + 8}\right)^n x^n.
\]
The root test for convergence involves analyzing the limit:
\[
\lim_{n \to \infty} \sqrt[n]{|a_n|}.
\]
By calculating this limit, you can determine the value of \(x\) for which the series is convergent. The radius of convergence \(R\) can then be found using the relation that the series converges if:
\[
\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1 \implies |x| < R.
\]
The challenge is to evaluate this limit and solve for \(R\).
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