4(-1)"e-n n = 1 converges diverges
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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Question
Test the series for convergence or divergence.
a)
b)
c)
![### Infinite Series Convergence Evaluation
In this section, we evaluate whether each of the following infinite series converges or diverges. For each series, choose whether it converges or diverges.
---
1. **Series 1:**
\[
\sum_{n=1}^{\infty} 4(-1)^n e^{-n}
\]
- ○ converges
- ○ diverges
---
2. **Series 2:**
\[
\sum_{n=1}^{\infty} (-1)^n \frac{4n^n}{n!}
\]
- ○ converges
- ○ diverges
---
3. **Series 3:**
\[
\sum_{n=1}^{\infty} (-1)^n \sin\left(\frac{5\pi}{n}\right)
\]
- ○ converges
- ○ diverges
---
Consider the behavior of each series as \( n \to \infty \) and use appropriate convergence tests, such as the Alternating Series Test, Ratio Test, or others, to determine the nature of each series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F885f4e27-5b29-4c6f-b568-f932f5320acb%2Fce2bc9d6-61d8-4db6-9dc8-0ed6f611aaa4%2F4o28xrp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Infinite Series Convergence Evaluation
In this section, we evaluate whether each of the following infinite series converges or diverges. For each series, choose whether it converges or diverges.
---
1. **Series 1:**
\[
\sum_{n=1}^{\infty} 4(-1)^n e^{-n}
\]
- ○ converges
- ○ diverges
---
2. **Series 2:**
\[
\sum_{n=1}^{\infty} (-1)^n \frac{4n^n}{n!}
\]
- ○ converges
- ○ diverges
---
3. **Series 3:**
\[
\sum_{n=1}^{\infty} (-1)^n \sin\left(\frac{5\pi}{n}\right)
\]
- ○ converges
- ○ diverges
---
Consider the behavior of each series as \( n \to \infty \) and use appropriate convergence tests, such as the Alternating Series Test, Ratio Test, or others, to determine the nature of each series.
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