https://file.aiquickdraw.com/mathbot/file/1730560837934sbv2ue1u.jpg

To determine whether the series 

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

is absolutely convergent or divergent using the Ratio Test, let's proceed with the following steps.

### Ratio Test Review
For a series \(\sum a_n\), the Ratio Test states:
- If \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L\),
  - If \(L < 1\), the series converges absolutely.
  - If \(L > 1\), the series diverges.
  - If \(L = 1\), the test is inconclusive.

In this case, the \(n\)-th term \(a_n\) of the series is:
\[
a_n = \frac{3x - 2}{n 3^n}.
\]

### Applying the Ratio Test
1. **Compute \(a_{n+1}\):**
   \[
   a_{n+1} = \frac{3x - 2}{(n+1) 3^{n+1}}.
   \]

2. **Calculate \(\left| \frac{a_{n+1}}{a_n} \right|\):**
   \[
   \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\frac{3x - 2}{(n+1) 3^{n+1}}}{\frac{3x - 2}{n 3^n}} \right| = \left| \frac{3x - 2}{3x - 2} \right| \cdot \frac{n}{n+1} \cdot \frac{1}{3}.
   \]
   Simplifying, we get:
   \[
   \left| \frac{a_{n+1}}{a_n} \right| = \frac{n}{n+1} \cdot \frac{1}{3}.
   \]

3. **Take the limit as \(n \to \infty\):**
   \[
   \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{n}{n+1} \cdot \frac{1}{3} = \frac{1}{3}.
   \]

### Conclusion
Since \(\frac{1}{3} < 1\), the series \(\sum_{n=1}^{\infty} \frac{3x - 2}{n 3^n}\) **converges absolutely** by the Ratio Test.

Would you like further details or have any questions on this solution?

---

### Related Questions
1. How does the Ratio Test compare to the Root Test for determining convergence?
2. Can the Ratio Test be used to determine the radius of convergence of a power series?
3. What would the outcome be if the limit was equal to 1 in the Ratio Test?
4. How would the series behave if the term \(3^n\) in the denominator were replaced with a different base, like \(4^n\)?
5. How does absolute convergence differ from conditional convergence in series?

### Tip
When using the Ratio Test, simplifying the terms before taking limits can make calculations more straightforward, especially for terms involving powers and factorials.

Determine whether the series is absolutely convergent or divergent using the ratio test for the series \( \sum_{n=1}^{\infty} \frac{3x - 2}{n 3^n} \).

Learn how to determine the absolute convergence or divergence of the series \( \sum_{n=1}^{\infty} \frac{3x - 2}{n 3^n} \) using the Ratio Test. This example involves calculating the limit of the ratio of consecutive terms to establish convergence.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation