The sequence in question is \( a_n = \frac{(-1)^n}{n} \). We want to determine whether this sequence converges, and if it does, find the limit.

### Step-by-Step Analysis:

1. **General Term**: The sequence is given by:
   \[
   a_n = \frac{(-1)^n}{n}
   \]
   Here, \((-1)^n\) alternates between \(-1\) and \(1\) depending on whether \(n\) is odd or even, respectively.

2. **Behavior of the Sequence**: 
   - When \(n\) is even: \( a_n = \frac{1}{n} \)
   - When \(n\) is odd: \( a_n = \frac{-1}{n} \)

   As \(n\) increases, the absolute value of \(a_n\) decreases because \( \frac{1}{n} \) tends to zero.

3. **Convergence**:
   - To determine the limit of \(a_n\) as \(n\) approaches infinity, observe that:
     \[
     \lim_{n \to \infty} \frac{1}{n} = 0 \quad \text{and} \quad \lim_{n \to \infty} \frac{-1}{n} = 0
     \]
   - Since both \( \frac{1}{n} \) and \( \frac{-1}{n} \) tend to 0 as \(n\) becomes large, regardless of whether \(n\) is odd or even, the entire sequence tends to 0.

4. **Conclusion**:
   The sequence converges to 0. In other words:
   \[
   \lim_{n \to \infty} a_n = 0
   \]

### Summary:
The sequence \( a_n = \frac{(-1)^n}{n} \) converges, and its limit is \(0\).

Would you like more details or have any questions?

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**Tip:** When analyzing sequences, it’s important to consider both the behavior as \(n\) increases and the nature of any alternating terms.

Explore the convergence of the sequence (-1)^n / n. Understand how alternating signs affect the limit of the sequence and learn about sequence convergence properties. Suitable for advanced undergraduate and graduate students.

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