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To determine the radius of convergence \( R \) for the series 

\[
\sum_{n=1}^{\infty} \frac{n^4 x^n}{17^n},
\]

we'll use the **Root Test** or **Ratio Test**.

### Step-by-Step Solution

1. **Rewrite the Series Terms**: 
   The general term of the series is:
   \[
   a_n = \frac{n^4 x^n}{17^n}.
   \]

2. **Apply the Ratio Test**:
   According to the Ratio Test, the radius of convergence \( R \) can be found by examining the limit:
   \[
   \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|.
   \]
   If this limit exists and equals \( \frac{1}{R} \), then we can solve for \( R \).

3. **Calculate \( \frac{a_{n+1}}{a_n} \)**:
   Substitute \( a_{n+1} = \frac{(n+1)^4 x^{n+1}}{17^{n+1}} \) and \( a_n = \frac{n^4 x^n}{17^n} \):
   \[
   \frac{a_{n+1}}{a_n} = \frac{(n+1)^4 x^{n+1} / 17^{n+1}}{n^4 x^n / 17^n} = \frac{(n+1)^4}{n^4} \cdot \frac{x}{17}.
   \]

4. **Take the Limit as \( n \to \infty \)**:
   \[
   \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left( \frac{(n+1)^4}{n^4} \cdot \frac{|x|}{17} \right).
   \]
   As \( n \to \infty \), \( \frac{(n+1)^4}{n^4} \to 1 \), so:
   \[
   \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{|x|}{17}.
   \]

5. **Set the Limit Less Than 1 for Convergence**:
   For convergence, we need:
   \[
   \frac{|x|}{17} < 1 \Rightarrow |x| < 17.
   \]

6. **Conclusion**:
   The radius of convergence \( R \) is therefore:
   \[
   R = 17.
   \]

### Answer
The correct choice is:
- **17**

Would you like more details, or do you have any questions about this solution?

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Here are some additional questions to explore:

1. What is the significance of the Ratio Test in determining the convergence of a series?
2. How would the radius of convergence change if the series had a different base, say \( 10^n \) instead of \( 17^n \)?
3. Can the Root Test also be applied to find the radius of convergence in this example?
4. What happens to the convergence of the series at the endpoints \( x = \pm 17 \)?
5. How does the term \( n^4 \) in the numerator affect the convergence properties of the series?

**Tip:** The Radius of Convergence can always be interpreted as the range of values for which the series converges around the center point \( x = 0 \).

Find the radius of convergence of the series: ∑(n=1 to ∞) (n^4 * x^n) / 17^n.

Calculate the radius of convergence for the series ∑(n=1 to ∞) (n^4 * x^n) / 17^n using the Ratio Test. This problem involves series and convergence concepts, suitable for undergraduate students.

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