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?

---

Related Questions:

1. What is the difference between pointwise convergence and uniform convergence?
2. How can we determine if a series converges using the comparison test?
3. What is the difference between absolute convergence and conditional convergence?
4. How do we prove that a sequence is bounded?
5. What is the Cauchy criterion for convergence of a sequence?
6. How does the squeeze theorem help in determining the limit of a sequence?
7. What is the relationship between convergence of a sequence and the convergence of its corresponding series?
8. How does the Alternating Series Test determine the convergence of a series?

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