Question

Determine whether the sequence is bounded, bounded above, bounded below, or none of the above. $$\left\\{a_{n}\right\\}=\left\\{(-1)^{n} \frac{3 n-1}{n}\right\\}$$

Short answer

The sequence is bounded both above and below.

Step-by-step solution

Identify Sequence Behavior

The sequence given is \( \{a_n\} = \{(-1)^n \frac{3n - 1}{n}\} \). Let's rewrite this expression as \( a_n = (-1)^n (3 - \frac{1}{n}) \), factoring \( \frac{1}{n} \) within the terms. This formula will help us analyze the behavior as \( n \) becomes very large.

Determine Even and Odd Terms

The sequence alternates sign due to the \((-1)^n\) term. For even \(n\), \((-1)^n = 1\), so the sequence terms are \( a_n = 3 - \frac{1}{n} \). For odd \( n \), \((-1)^n = -1\), and the terms become \( a_n = -3 + \frac{1}{n} \). This shows the sequence switches between being slightly below 3 and slightly above -3 as \( n \) increases.

Analyze Boundedness

For even \( n \), as \( n \to \infty \), the terms approach 3 from below (i.e., \( 3 - \frac{1}{n} < 3 \)). For odd \( n \), as \( n \to \infty \), the terms approach -3 from above (i.e., \(-3 + \frac{1}{n} > -3 \)). Therefore, all terms lie within the interval \((-3, 3)\), indicating the sequence is bounded.

Conclusion on Bounds

Since the terms are confined within \(-3\) and \(3\), the sequence is both bounded above by 3 and bounded below by -3. The entire behavior of the sequence conforms within these constant boundaries, regardless of the oscillating pattern introduced by \((-1)^n\).

Key concepts

sequence behavior

When analyzing sequence behavior, we are interested in understanding how the sequence evolves as the number of terms increases. In this particular sequence, given by \( a_n = (-1)^n (3 - \frac{1}{n}) \), the behavior is influenced by two parts: the alternating sign, \((-1)^n\), and the terms \(3 - \frac{1}{n}\).
This sequence is a mix of predictable changes: one due to the sign alternating between positive and negative as \(n\) changes from even to odd, and the other due to the term \(\frac{1}{n}\) becoming very small as \(n\) becomes large.

• For even \(n\), \(a_n\) approaches \(3\) as it becomes larger, showing the behavior of approaching a stable value.
• For odd \(n\), the terms inch closer to \(-3\), again indicating a consistent narrowing toward a particular range.

Understanding this behavior is crucial because it reveals that the sequence, despite its alternating terms, converges toward boundaries without divergence, remaining in a specific interval. This is a fundamental aspect when discussing the boundedness of sequences.

alternating sequences

Alternating sequences, like the one presented, have a unique characteristic of switching signs with each subsequent term, influenced by the term \((-1)^n\). This creates a pattern where each term flips from positive to negative as we progress through the sequence.
Such sequences can be particularly intriguing because:

• They create a back-and-forth movement, or oscillation, that might imply instability at first glance.
• However, in more controlled sequences, like our example, this oscillation doesn't lead to unboundedness.

For instance, in our sequence:
- When \(n\) is even, \(a_n\) becomes positive and approaches 3
- When \(n\) is odd, \(a_n\) becomes negative and approaches -3
So despite showing an alternating nature, this sequence remains within the confines of a predictable boundary. This controlled alternation highlights the importance of understanding each component influencing the sequence's behavior, reinforcing the necessity to consider all variables in sequence analysis.

limits and bounds analysis

The analysis of limits and bounds of a sequence is essential to determine if a sequence is bounded and within what range. We utilize the concept of limits to explore how the terms behave as \(n\) tends to infinity.
In the provided sequence, for both even and odd \(n\), the terms \(3 - \frac{1}{n}\) and \(-3 + \frac{1}{n}\) help us find limits:

• As \(n\rightarrow\infty\), \(\frac{1}{n}\) approaches zero.
• For even \(n\), \(a_n\) approaches 3 from below. Thus,
  - Upper bound: 3
• For odd \(n\), \(a_n\) approaches -3 from above. Hence,
  - Lower bound: -3

This thorough analysis using limits reveals the bounded nature of the sequence. Since all terms remain between -3 and 3, the sequence is confined within clear bounds. Understanding these bounds is key to grasping how sequences stay within certain parameters. They give us the exact range in which the sequence lies, offering clarity on its behavior over infinite terms.