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The problem you provided asks to determine whether the series

\[
\sum_{n=1}^{\infty} \frac{n(x-4)^n}{3^n}
\]

is absolutely convergent or divergent using the ratio test.

### Solution
To apply the ratio test, we consider the absolute value of the general term of the series, \( a_n = \frac{n |x-4|^n}{3^n} \).

1. **Ratio Test Setup:**
   The ratio test requires calculating the limit:
   \[
   L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
   \]
   where \( a_n = \frac{n |x-4|^n}{3^n} \).

2. **Compute \( a_{n+1} \):**
   \[
   a_{n+1} = \frac{(n+1) |x-4|^{n+1}}{3^{n+1}} = \frac{(n+1) |x-4|^{n+1}}{3 \cdot 3^n}
   \]

3. **Set Up the Ratio:**
   \[
   \frac{a_{n+1}}{a_n} = \frac{(n+1) |x-4|^{n+1}}{3 \cdot 3^n} \cdot \frac{3^n}{n |x-4|^n} = \frac{(n+1) |x-4|}{3n}
   \]

4. **Take the Limit as \( n \to \infty \):**
   \[
   L = \lim_{n \to \infty} \frac{(n+1) |x-4|}{3n} = \frac{|x-4|}{3} \lim_{n \to \infty} \frac{n+1}{n} = \frac{|x-4|}{3}
   \]

5. **Apply the Ratio Test Conclusion:**
   The ratio test states that:
   - If \( L < 1 \), the series converges absolutely.
   - If \( L > 1 \), the series diverges.
   - If \( L = 1 \), the test is inconclusive.

   Therefore, the series will converge absolutely if
   \[
   \frac{|x-4|}{3} < 1
   \]
   which simplifies to
   \[
   |x-4| < 3
   \]

6. **Solution in Terms of \( x \):**
   This inequality implies:
   \[
   -3 < x - 4 < 3
   \]
   or
   \[
   1 < x < 7
   \]

### Conclusion
The series converges absolutely for \( x \) in the interval \( (1, 7) \).

Would you like further details or have any questions?

---

Here are 5 related questions to expand your understanding:

1. How would the ratio test differ if the series included a factorial in the numerator or denominator?
2. Can the ratio test determine convergence if \( L = 1 \)?
3. How does absolute convergence differ from conditional convergence?
4. What would happen if the series was \(\sum_{n=1}^{\infty} \frac{n^2 (x-4)^n}{3^n}\)?
5. Why is it helpful to check for absolute convergence when dealing with power series?

**Tip:** The ratio test is particularly useful for series with terms involving powers, as it simplifies with ratios.

Determine whether the series is absolutely convergent or divergent using the ratio test:

\[
\sum_{n=1}^{\infty} \frac{n(x-4)^n}{3^n}
\]

Learn to determine if the series \( \sum_{n=1}^{\infty} \frac{n(x-4)^n}{3^n} \) is absolutely convergent or divergent using the ratio test. This solution involves applying the ratio test to find the interval of convergence based on the limit. Suitable for high school students in Grades 11-12.

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