-Show how the series converges absolutely COS s (3n + 1) n5 +4 N3D1
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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![**Problem 22**: Show how the series converges absolutely
\[
\sum_{n=1}^{\infty} \frac{\cos(3n + 1)}{n^5 + 4}
\]
**Explanation**:
This problem involves demonstrating the absolute convergence of the given infinite series. Absolute convergence occurs if the series of absolute values,
\[
\sum_{n=1}^{\infty} \left|\frac{\cos(3n + 1)}{n^5 + 4}\right|
\]
converges. To determine this, you can use various tests such as the Comparison Test, the Ratio Test, or the Root Test. The behavior of \(\cos(3n + 1)\) is bounded as \(-1 \leq \cos(3n + 1) \leq 1\), so the series can be effectively compared with a simpler series for convergence analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2396c550-bba7-4086-87a5-883b238a150c%2F0b7916d1-5c74-414c-ad52-68b6cc149837%2Fsyxydq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 22**: Show how the series converges absolutely
\[
\sum_{n=1}^{\infty} \frac{\cos(3n + 1)}{n^5 + 4}
\]
**Explanation**:
This problem involves demonstrating the absolute convergence of the given infinite series. Absolute convergence occurs if the series of absolute values,
\[
\sum_{n=1}^{\infty} \left|\frac{\cos(3n + 1)}{n^5 + 4}\right|
\]
converges. To determine this, you can use various tests such as the Comparison Test, the Ratio Test, or the Root Test. The behavior of \(\cos(3n + 1)\) is bounded as \(-1 \leq \cos(3n + 1) \leq 1\), so the series can be effectively compared with a simpler series for convergence analysis.
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