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To determine the radius and interval of convergence for the given series

\[
\sum_{n=1}^{\infty} \frac{3x - 2}{n 3^n},
\]

we can apply the **Ratio Test**. The Ratio Test states that for a series \(\sum a_n\), if

\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L,
\]

then:
1. The series converges if \(L < 1\),
2. The series diverges if \(L > 1\),
3. The test is inconclusive if \(L = 1\).

### Step-by-Step Solution

1. **Identify \(a_n\):**
   For the given series, the general term is:
   \[
   a_n = \frac{3x - 2}{n 3^n}.
   \]

2. **Compute \(\left| \frac{a_{n+1}}{a_n} \right|\):**
   We need to find \(a_{n+1}\) and then set up the ratio:
   \[
   a_{n+1} = \frac{3x - 2}{(n+1) 3^{n+1}}.
   \]
   Now, we calculate the ratio:
   \[
   \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\frac{3x - 2}{(n+1) 3^{n+1}}}{\frac{3x - 2}{n 3^n}} \right|.
   \]

3. **Simplify the Ratio:**
   This simplifies to:
   \[
   \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{3x - 2}{3} \cdot \frac{n}{n+1} \right|.
   \]
   Taking the limit as \(n \to \infty\), we get:
   \[
   \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{3x - 2}{3} \right|.
   \]

4. **Set up the inequality for convergence:**
   For convergence, we require:
   \[
   \left| \frac{3x - 2}{3} \right| < 1.
   \]
   Simplifying this inequality, we find:
   \[
   -1 < \frac{3x - 2}{3} < 1.
   \]
   Multiply through by 3:
   \[
   -3 < 3x - 2 < 3.
   \]
   Adding 2 to each part:
   \[
   -1 < 3x < 5.
   \]
   Now, divide by 3:
   \[
   -\frac{1}{3} < x < \frac{5}{3}.
   \]

5. **Conclusion:**
   The **radius of convergence** is:
   \[
   R = \frac{5}{3} - \left(-\frac{1}{3}\right) = \frac{2}{3}.
   \]
   The **interval of convergence** is:
   \[


Find the radius of convergence and the interval of convergence of the given series using the ratio test: ∑(3x - 2) / (n * 3^n) from n=1 to infinity.

Calculate the radius and interval of convergence for the series ∑(3x - 2) / (n * 3^n) using the Ratio Test. This solution provides step-by-step instructions, including setting up the Ratio Test, finding limits, and solving inequalities to determine convergence range.

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