prove that the sequence converge {1-(-1)^n/n}

We are tasked with proving that the sequence \( a_n = 1 - \frac{(-1)^n}{n} \) converges.

### Step 1: Understanding the sequence

The sequence is defined as:
\[
a_n = 1 - \frac{(-1)^n}{n}.
\]
Here, \( (-1)^n \) alternates between \( 1 \) and \( -1 \) depending on whether \( n \) is even or odd. Therefore, we can break this into two subsequences based on whether \( n \) is even or odd:

- When \( n \) is even, \( (-1)^n = 1 \), so the sequence becomes:
  \[
  a_n = 1 - \frac{1}{n}.
  \]
- When \( n \) is odd, \( (-1)^n = -1 \), so the sequence becomes:
  \[
  a_n = 1 - \frac{-1}{n} = 1 + \frac{1}{n}.
  \]

### Step 2: Analyze the behavior as \( n \to \infty \)

Now let's analyze the behavior of these subsequences as \( n \) becomes very large.

- When \( n \) is even: \( a_n = 1 - \frac{1}{n} \to 1 \) as \( n \to \infty \).
- When \( n \) is odd: \( a_n = 1 + \frac{1}{n} \to 1 \) as \( n \to \infty \).

In both cases, the value of \( a_n \) approaches 1 as \( n \to \infty \), regardless of whether \( n \) is even or odd.

### Step 3: Conclusion

Since both subsequences converge to the same limit, we conclude that the sequence \( a_n = 1 - \frac{(-1)^n}{n} \) converges to 1.

Thus, the sequence converges, and:
\[
\lim_{n \to \infty} \left( 1 - \frac{(-1)^n}{n} \right) = 1.
\]

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Would you like more details on any step or have any further questions?

Here are five related questions:
1. How do we define the limit of a sequence?
2. Can a sequence converge to different limits based on subsequences?
3. What is the definition of a convergent sequence?
4. How do alternating sequences behave in terms of convergence?
5. How would the sequence change if the term \( (-1)^n \) were replaced with another function?

**Tip**: Breaking a sequence into subsequences can simplify the analysis of convergence, especially for alternating or periodic terms.

Learn how to prove that the sequence a_n = 1 - (-1)^n/n converges as n approaches infinity. This step-by-step solution covers key concepts such as limits and alternating sequences, making it suitable for advanced high school or undergraduate students.

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