https://file.aiquickdraw.com/mathbot/file/1733461701903ecqbpygn.jpg

The uploaded image involves solving a problem related to the *radius of convergence* of a series and the *interval of convergence*. Here's a breakdown of what the steps show and their significance:

### Observations:
1. **Test for Convergence**:
   The user is determining the radius of convergence using the *Ratio Test*, which involves:
   - Comparing the ratio \(\frac{a_{n+1}}{a_n}\), simplifying it, and taking the limit as \(n \to \infty\).

2. **Steps Illustrated**:
   - The ratio \(\frac{|a_{n+1}|}{|a_n|}\) is computed.
   - There’s simplification of expressions involving \(x^n\) and factorials \(n!\).
   - The final limit is calculated to identify the boundary of convergence.

3. **Key Concept**:
   The series converges if the limit from the ratio test satisfies \(|x| < R\), where \(R\) is the radius of convergence. Beyond this, further checks (like endpoints) would be needed for the interval of convergence.

Would you like me to:
- **Verify the calculations in detail**?
- **Provide explanations for each step shown in the image**?

### Follow-up Questions:
1. How does the Ratio Test work, and why is it suitable here?
2. What is the importance of simplifying factorial terms in the limit?
3. How do we test the interval of convergence after determining the radius?
4. Can the Alternating Series Test be applied here for endpoint behavior?
5. Why do we often use limits as \(n \to \infty\) in convergence tests?

### Tip:
For series involving factorials or exponential terms, always simplify the ratio by dividing common factors and expanding expressions like \(n+1\) to cancel terms efficiently.

What is the radius and interval of convergence for the given series? Use the ratio test to determine the solution.

Step-by-step guide to finding the radius and interval of convergence for a series using the ratio test. Includes detailed calculations and simplifications for series involving factorials. Ideal for undergraduate math students.

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