O Determine whether E=1 converges or diverges. n! 104n t.
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
Justify answer by citing a relevant test.
![**Determine whether the series**
\[
\sum_{n=1}^{\infty} \frac{n!}{10^{4n}}
\]
**converges or diverges.**
*Instructions for Analysis:*
1. **Understand the Terms:**
- \(n!\) denotes the factorial of \(n\), meaning the product of all positive integers up to \(n\).
- \(10^{4n}\) is an exponential expression, where the base 10 is raised to the power of \(4n\).
2. **Approach to Determine Convergence or Divergence:**
- Consider using tests for convergence such as the Ratio Test or Comparison Test.
- Analyze the growth of the factorial in the numerator compared to the exponential growth in the denominator.
By following these steps, students should be able to ascertain whether this infinite series converges or diverges.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe872339c-719c-4a81-bc7c-37653a99cb41%2F61983b98-cc4a-4ece-ae39-d09dc5147682%2F77govfx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine whether the series**
\[
\sum_{n=1}^{\infty} \frac{n!}{10^{4n}}
\]
**converges or diverges.**
*Instructions for Analysis:*
1. **Understand the Terms:**
- \(n!\) denotes the factorial of \(n\), meaning the product of all positive integers up to \(n\).
- \(10^{4n}\) is an exponential expression, where the base 10 is raised to the power of \(4n\).
2. **Approach to Determine Convergence or Divergence:**
- Consider using tests for convergence such as the Ratio Test or Comparison Test.
- Analyze the growth of the factorial in the numerator compared to the exponential growth in the denominator.
By following these steps, students should be able to ascertain whether this infinite series converges or diverges.
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