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To find the radius and interval of convergence for the given series using the Ratio Test, we have the series:

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

To apply the Ratio Test, we examine the general term of the series:
\[
a_n = \frac{3x - 2}{n \cdot 3^n}
\]

### Step 1: Applying the Ratio Test
The Ratio Test states that for a series \(\sum a_n\), if we compute the limit
\[
L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
\]
and find that \(L < 1\), then the series converges absolutely. If \(L > 1\), the series diverges. If \(L = 1\), the test is inconclusive.

Let's find \(\frac{a_{n+1}}{a_n}\):

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

2. **Form the ratio \(\left| \frac{a_{n+1}}{a_n} \right|\):**
   \[
   \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\frac{3x - 2}{(n+1) \cdot 3^{n+1}}}{\frac{3x - 2}{n \cdot 3^n}} \right|
   \]
   Simplifying this expression:
   \[
   = \left| \frac{3x - 2}{3x - 2} \cdot \frac{n}{(n+1) \cdot 3} \right|
   \]
   \[
   = \frac{n}{3(n+1)}
   \]

3. **Take the limit as \(n \to \infty\):**
   \[
   L = \lim_{n \to \infty} \frac{n}{3(n+1)} = \lim_{n \to \infty} \frac{n}{3n + 3} = \frac{1}{3}
   \]

Since \(L = \frac{1}{3} < 1\), the Ratio Test tells us that the series converges for all values of \(x\).

### Conclusion
- **Radius of Convergence:** Since the series converges for all \(x\), the radius of convergence \(R\) is \(\infty\).
- **Interval of Convergence:** The interval of convergence is \((-\infty, \infty)\).

Would you like a deeper explanation of any of these steps?

### Related Questions
1. What does it mean for a series to converge absolutely?
2. How would the Ratio Test be applied if the series had a factor of \(x^n\) instead?
3. What is the importance of the Ratio Test in determining convergence?
4. How does the root test compare with the ratio test for finding convergence?
5. Why does a radius of convergence of \(\infty\) imply convergence for all real numbers?

**Tip:** The Ratio Test is particularly useful for series involving factorials or exponential terms, as these lead to clear limits when ratios of successive terms are computed.

Find the radius of convergence and the interval of convergence of the given series using the ratio test:

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

Explore the step-by-step solution for finding the radius and interval of convergence of the series \( \sum_{n=1}^{\infty} \frac{3x - 2}{n \cdot 3^n} \) using the Ratio Test. This guide covers essential calculus concepts related to series convergence, radius of convergence, and interval of convergence.

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