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To solve this problem, we need to determine the radius of convergence \( R \) and the interval of convergence \( I \) for the given power series:

\[
\sum_{n=1}^{\infty} \frac{n^6 (x - 16)^n}{2 \cdot 4 \cdot 6 \cdots (2n)}.
\]

### Step 1: Determine the Radius of Convergence \( R \)

1. First, let's observe that the series has the form:
   \[
   \sum_{n=1}^{\infty} a_n (x - 16)^n,
   \]
   where \( a_n = \frac{n^6}{2 \cdot 4 \cdot 6 \cdots (2n)} \).

2. To find the radius of convergence, we can use the **Ratio Test**. According to the Ratio Test, for a series \( \sum a_n (x - 16)^n \), the radius of convergence \( R \) is given by:
   \[
   \frac{1}{R} = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|.
   \]

3. We need to find an expression for \( \frac{a_{n+1}}{a_n} \) using the given terms for \( a_n \).

4. Simplifying the factorial terms in \( a_n \), we find that this series has a radius of convergence \( R = 1 \).

### Step 2: Determine the Interval of Convergence \( I \)

1. With \( R = 1 \), the interval of convergence is centered at \( x = 16 \), meaning we test the interval \( x \in (16 - 1, 16 + 1) \) or \( x \in (15, 17) \).

2. **Check Endpoints** \( x = 15 \) and \( x = 17 \) by substituting into the series to see if convergence holds.

   - **At \( x = 15 \):** Substitute \( x = 15 \) in the series, which becomes an alternating series, and it converges.
   - **At \( x = 17 \):** Substitute \( x = 17 \) in the series, and this series converges as well.

Thus, the interval of convergence \( I \) is \([15, 17]\).

### Final Answer

- **Radius of Convergence, \( R \):** 1
- **Interval of Convergence, \( I \):** \([15, 17]\)

Would you like further explanation on any step?

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#### Further Questions
1. What is the Ratio Test, and how is it applied in finding convergence?
2. How can factorial expressions in a series term affect convergence?
3. Why do we test the endpoints separately when finding the interval of convergence?
4. How does the presence of \( n^6 \) in the numerator influence the convergence?
5. Can the Radius of Convergence ever be negative?

#### Tip
The Ratio Test is often the quickest way to determine the radius of convergence for series with factorial terms.

Given the power series \( \sum_{n=1}^{\infty} \frac{n^6 (x - 16)^n}{2 \cdot 4 \cdot 6 \cdots (2n)} \). Find the radius of convergence, \( R \), and the interval of convergence, \( I \).

Learn how to determine the radius of convergence and interval of convergence for the power series \( \sum_{n=1}^{\infty} \frac{n^6 (x - 16)^n}{2 \cdot 4 \cdot 6 \cdots (2n)} \). This solution explores the Ratio Test and techniques for testing endpoints to confirm the interval \( [15, 17] \).

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