https://file.aiquickdraw.com/mathbot/file/1730168169096tlofdnma.jpg

Based on your screenshot, it seems that the question asks you to use the root test to determine the convergence of the series with the general term:

\[
\left( \frac{3n + 2}{9n + 1} \right)^{2n}
\]

You entered the following values:
- **Limit calculation**: \( \frac{1}{3} \)
- **Comparison**: "less than 1"
- **Conclusion**: "converges"

Let's break down the solution and check why some of your answers were marked incorrect:

1. **Root Test**: The root test states that for a series \(\sum a_n\), if we consider \(\lim_{n \to \infty} \sqrt[n]{|a_n|}\):
   - If this limit is less than 1, the series **converges** absolutely.
   - If this limit is greater than 1, the series **diverges**.
   - If this limit equals 1, the test is **inconclusive**.

2. **Finding the Limit**:
   - The term given is \( \left( \frac{3n + 2}{9n + 1} \right)^{2n} \).
   - Taking the \(n\)th root, we have:
     \[
     \sqrt[n]{\left( \frac{3n + 2}{9n + 1} \right)^{2n}} = \left( \frac{3n + 2}{9n + 1} \right)^2
     \]
   - Now, calculate \(\lim_{n \to \infty} \left( \frac{3n + 2}{9n + 1} \right)^2\):
     - Simplifying \(\frac{3n + 2}{9n + 1}\) as \(n \to \infty\), we get \(\frac{3}{9} = \frac{1}{3}\).
     - Thus, \(\lim_{n \to \infty} \left( \frac{3n + 2}{9n + 1} \right)^2 = \left( \frac{1}{3} \right)^2 = \frac{1}{9}\).

3. **Conclusion**:
   - Since \(\frac{1}{9} < 1\), the series **converges** by the root test.

### Corrections
- The correct limit is \(\frac{1}{9}\) rather than \(\frac{1}{3}\).
- Your conclusion that the series converges was correct, but updating the limit to \(\frac{1}{9}\) should ensure full credit.

Would you like a detailed explanation of each step or further assistance?

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### Related Questions:
1. What other convergence tests could be used for series with terms raised to the power \(2n\)?
2. How does the ratio test compare with the root test for series with exponential terms?
3. Why does the limit approaching a value less than 1 imply convergence in the root test?
4. Could there be an error in the convergence if a term with \(n\)th power behaves differently?
5. When is the root test considered inconclusive?

### Tip:
Always simplify the expression within the root to avoid calculation errors when applying the root test.

Use the root test to determine whether the series \( \sum \left( \frac{3n + 2}{9n + 1} \right)^{2n} \) converges or diverges.

Apply the root test to determine the convergence of the series \( \sum \left( \frac{3n + 2}{9n + 1} \right)^{2n} \). This solution includes detailed steps, formula breakdown, and final conclusion on convergence.

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