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The given sequence is:

\[
a_n = \frac{\cos^2(n)}{3^n}
\]

We are asked to determine whether the sequence converges or diverges, and if it converges, to find the limit.

### Step-by-Step Solution:

1. **Behavior of the numerator**: The function \(\cos^2(n)\) oscillates between 0 and 1 for all values of \(n\). Specifically, since \(\cos(n)\) is periodic and bounded, \(\cos^2(n)\) is also bounded, i.e.,
   \[
   0 \leq \cos^2(n) \leq 1 \quad \text{for all } n.
   \]

2. **Behavior of the denominator**: The term \(3^n\) grows exponentially as \(n\) increases, since \(3^n\) represents an exponentially increasing function with base 3.

3. **Limit of the sequence**: As \(n \to \infty\), the exponential growth of \(3^n\) in the denominator will dominate the oscillating \(\cos^2(n)\) in the numerator. This leads to the conclusion that the entire sequence tends to 0, because the denominator grows much faster than the numerator can oscillate.

   Formally, we evaluate the limit:
   \[
   \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{\cos^2(n)}{3^n} = 0.
   \]

### Conclusion:

The sequence **converges** and the limit is:

\[
\lim_{n \to \infty} a_n = 0.
\]

Would you like further details or clarifications on any step? 

Here are 5 related questions to expand your understanding:
1. What are some other sequences where the exponential term dominates the numerator?
2. How would the limit change if the denominator were \(2^n\) instead of \(3^n\)?
3. Can a sequence with a trigonometric function in the numerator ever diverge?
4. How does the periodic nature of \(\cos(n)\) affect the convergence of sequences in general?
5. What is the general approach for determining the limit of sequences involving exponential growth?

**Tip**: When dealing with sequences involving exponential terms, check if the exponential growth in the denominator can overpower any bounded or oscillating terms in the numerator to determine convergence.

Determine whether the sequence converges or diverges. If it converges, find the limit: a_n = cos^2(n) / 3^n.

Determine if the sequence a_n = cos^2(n) / 3^n converges or diverges. This problem explores the behavior of oscillating and exponentially growing sequences, leading to the conclusion that the sequence converges to 0. Suitable for students in Grades 11-12.

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