Consider the following sequences, and determine if they converge or not in Q.
1. (2)n∈N
2. 
2 − 1
n

n∈N

Let's examine the given sequences to determine whether they converge in the set of rational numbers, \( \mathbb{Q} \).

### 1. Sequence: \( (2)_{n \in \mathbb{N}} \)

This sequence is constant, meaning that for all natural numbers \( n \), the term in the sequence is \( 2 \). So, the sequence is:

\[
2, 2, 2, 2, \dots
\]

Since every term of the sequence is the same (i.e., 2), the sequence trivially converges to \( 2 \) in \( \mathbb{Q} \), as it stays fixed at the value 2 for all \( n \).

**Conclusion**: This sequence converges to \( 2 \) in \( \mathbb{Q} \).

### 2. Sequence: \( \left( 2 - \frac{1}{n} \right)_{n \in \mathbb{N}} \)

The general term of this sequence is:

\[
a_n = 2 - \frac{1}{n}
\]

As \( n \to \infty \), the fraction \( \frac{1}{n} \) tends to 0. Therefore, the sequence tends to:

\[
\lim_{n \to \infty} a_n = 2 - \lim_{n \to \infty} \frac{1}{n} = 2 - 0 = 2
\]

Hence, this sequence converges to \( 2 \) as \( n \to \infty \).

**Conclusion**: This sequence also converges to \( 2 \) in \( \mathbb{Q} \).

### Final Conclusion:
Both sequences converge in \( \mathbb{Q} \), and both converge to the value 2.

---

Do you have any questions or would you like more details?

Here are 5 related questions:
1. What does it mean for a sequence to converge in \( \mathbb{Q} \)?
2. Can you provide an example of a sequence that does not converge in \( \mathbb{Q} \)?
3. How does convergence in \( \mathbb{R} \) differ from convergence in \( \mathbb{Q} \)?
4. How do we determine the limit of a sequence formally using the definition of a limit?
5. Can a constant sequence converge to a different value other than the constant itself?

**Tip**: A sequence is said to converge if, as \( n \) becomes large, the terms of the sequence approach a single value (the limit).

Consider the following sequences, and determine if they converge or not in Q. 1. (2)n∈N 2. 2 − 1/n n∈N

Determine if the sequences (2)n∈N and (2 − 1/n)n∈N converge in the set of rational numbers (Q). Learn about sequence convergence, limits, and rational number sets with step-by-step explanations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation