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?

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.