Find the radius of convergence, R, of the series. x2n n- 2 n (In(n))ª R = 1 Find the interval, I, of convergence of the series. (Enter your answer using interval notation.) I = (-1,1)
Find the radius of convergence, R, of the series. x2n n- 2 n (In(n))ª R = 1 Find the interval, I, of convergence of the series. (Enter your answer using interval notation.) I = (-1,1)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Text:**
Find the radius of convergence, \( R \), of the series.
\[
\sum_{n=2}^{\infty} \frac{x^{2n}}{n (\ln(n))^4}
\]
\[ R = \boxed{1} \; \checkmark \]
Find the interval, \( I \), of convergence of the series. (Enter your answer using interval notation.)
\[ I = \boxed{(-1,1)} \; \times \]
**Explanation:**
The equation shown is an infinite series where the terms are determined by:
\[
\frac{x^{2n}}{n (\ln(n))^4}
\]
The problem asks for two solutions:
1. **Radius of Convergence (\( R \)):** The series has a radius of convergence \( R = 1 \), which is verified with a checkmark.
2. **Interval of Convergence (\( I \)):** Initially selected as \((-1, 1)\), this interval is marked incorrect with a cross.
In mathematical analysis, the radius of convergence \( R \) helps determine for which values of \( x \) the series converges. For the interval of convergence, additional rules, such as testing endpoints, are often considered to establish where the series converges absolutely, conditionally, or diverges.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb192230a-9a5f-4cd4-8831-be6be5650fda%2Fc9ede163-175c-4cd2-b651-09bbf48f2ce1%2Flt4zs2s_processed.png&w=3840&q=75)
Transcribed Image Text:**Text:**
Find the radius of convergence, \( R \), of the series.
\[
\sum_{n=2}^{\infty} \frac{x^{2n}}{n (\ln(n))^4}
\]
\[ R = \boxed{1} \; \checkmark \]
Find the interval, \( I \), of convergence of the series. (Enter your answer using interval notation.)
\[ I = \boxed{(-1,1)} \; \times \]
**Explanation:**
The equation shown is an infinite series where the terms are determined by:
\[
\frac{x^{2n}}{n (\ln(n))^4}
\]
The problem asks for two solutions:
1. **Radius of Convergence (\( R \)):** The series has a radius of convergence \( R = 1 \), which is verified with a checkmark.
2. **Interval of Convergence (\( I \)):** Initially selected as \((-1, 1)\), this interval is marked incorrect with a cross.
In mathematical analysis, the radius of convergence \( R \) helps determine for which values of \( x \) the series converges. For the interval of convergence, additional rules, such as testing endpoints, are often considered to establish where the series converges absolutely, conditionally, or diverges.
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