Test the series below for convergence using the Ratio Test. (-1) "10²+1 (2n + 1)! n=0 The limit of the ratio test simplifies to lim f(n)| where f(n) = The limit is: 0 (enter oo for infinity if needed) Based on this, the series Converges Question Help: Message instructor Add Work
Test the series below for convergence using the Ratio Test. (-1) "10²+1 (2n + 1)! n=0 The limit of the ratio test simplifies to lim f(n)| where f(n) = The limit is: 0 (enter oo for infinity if needed) Based on this, the series Converges Question Help: Message instructor Add Work
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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![### Ratio Test for Series Convergence
**Series to Test:**
\[ \sum_{n=0}^{\infty} \frac{(-1)^n 10^{2n+1}}{(2n+1)!} \]
**Procedure Using the Ratio Test:**
1. Identify \( f(n) \) and the limit you need to evaluate.
2. Compute the limit \( \lim_{n \to \infty} \left| \frac{f(n+1)}{f(n)} \right| \).
**Form of \( f(n) \):**
The function \( f(n) \) here represents each term in the series:
\[ f(n) = \frac{(-1)^n 10^{2n+1}}{(2n+1)!} \]
**Simplifying the Limit:**
\[ \lim_{n \to \infty} \left| \frac{f(n+1)}{f(n)} \right| \]
**Limit Calculation:**
Enter the value of the limit \( \frac{f(n+1)}{f(n)} \) which we are testing for convergence. In specific problems like this, you often compute it step-by-step:
\[ \lim_{n \to \infty} \left| \frac{\frac{(-1)^{n+1} 10^{2(n+1)+1}}{(2(n+1)+1)!}}{\frac{(-1)^n 10^{2n+1}}{(2n+1)!}} \right| \]
Here you would then enter the calculated limit:
\[ \text{The limit is: } \boxed{0} \]
**Conclusion Based on Limit:**
Finally, based on the ratio test, state whether the series converges or diverges. For the given limit:
\[ \boxed{0} \]
The series:
\[ \boxed{\text{Converges}} \]
**Support and Further Help:**
For additional questions or help, you can:
- **Message Instructor:** Get in touch with your instructor directly for personalized assistance.
- **Add Work:** Include your calculations or steps you followed for a detailed review.
*Note: If there were any graphs or diagrams, they should be explained here, but this specific example does not include any graphical elements.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F41a4f8e2-50b1-40c5-ab3a-6358b729bd6f%2Fa76661c1-77bf-457d-856d-f3b9b7995555%2Fxrtm208_processed.png&w=3840&q=75)
Transcribed Image Text:### Ratio Test for Series Convergence
**Series to Test:**
\[ \sum_{n=0}^{\infty} \frac{(-1)^n 10^{2n+1}}{(2n+1)!} \]
**Procedure Using the Ratio Test:**
1. Identify \( f(n) \) and the limit you need to evaluate.
2. Compute the limit \( \lim_{n \to \infty} \left| \frac{f(n+1)}{f(n)} \right| \).
**Form of \( f(n) \):**
The function \( f(n) \) here represents each term in the series:
\[ f(n) = \frac{(-1)^n 10^{2n+1}}{(2n+1)!} \]
**Simplifying the Limit:**
\[ \lim_{n \to \infty} \left| \frac{f(n+1)}{f(n)} \right| \]
**Limit Calculation:**
Enter the value of the limit \( \frac{f(n+1)}{f(n)} \) which we are testing for convergence. In specific problems like this, you often compute it step-by-step:
\[ \lim_{n \to \infty} \left| \frac{\frac{(-1)^{n+1} 10^{2(n+1)+1}}{(2(n+1)+1)!}}{\frac{(-1)^n 10^{2n+1}}{(2n+1)!}} \right| \]
Here you would then enter the calculated limit:
\[ \text{The limit is: } \boxed{0} \]
**Conclusion Based on Limit:**
Finally, based on the ratio test, state whether the series converges or diverges. For the given limit:
\[ \boxed{0} \]
The series:
\[ \boxed{\text{Converges}} \]
**Support and Further Help:**
For additional questions or help, you can:
- **Message Instructor:** Get in touch with your instructor directly for personalized assistance.
- **Add Work:** Include your calculations or steps you followed for a detailed review.
*Note: If there were any graphs or diagrams, they should be explained here, but this specific example does not include any graphical elements.*
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