Chapter 8: Problem 32
Determine whether the sequence is bounded, bounded above, bounded below, or none of the above. $$\left\\{a_{n}\right\\}=\left\\{\frac{3 n^{2}-1}{n}\right\\}$$
Short Answer
Expert verified
The sequence is not bounded above or below.
Step by step solution
01
Find the General Term
Identify the formula for the terms of the sequence. The given sequence is \[ a_n = \frac{3n^2 - 1}{n} \]
02
Simplify the Formula
Simplify the expression for the sequence terms to make it easier to analyze. The expression can be simplified as follows:\[ a_n = \frac{3n^2 - 1}{n} = 3n - \frac{1}{n} \]
03
Determine the Behavior of the Sequence
Analyze the behavior of the sequence as \( n \to \infty \). As \( n \) becomes very large, \( \frac{1}{n} \) approaches zero. Thus,\[ a_n = 3n - \frac{1}{n} \approx 3n \]indicates that \( a_n \) increases without bound.
04
Evaluate Boundedness
To determine if the sequence is bounded above or below, check the growth of \( a_n \). Since \( a_n \rightarrow \infty \) as \( n \to \infty \), it is not bounded above. Evaluate these characteristics for lower bounds: for any positive \( n \), \( a_n = 3n - \frac{1}{n} \) shows even for smallest \( n \), it approaches zero; thus sequence can be negative for small \( n \). Hence, not bounded below.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sequence Behavior
In mathematics, understanding sequence behavior is crucial for analyzing how sequences progress. A sequence is essentially a list of numbers in a specific order. Each number in this list can be expressed in terms of an index, often denoted as \( n \), which tells us the position of the term in the sequence.
To comprehend the sequence's behavior, observe how each term is affected as we keep increasing \( n \). For the given sequence \( a_n = \frac{3n^2 - 1}{n} \), simplifying helps reveal its character more clearly.
On simplification, we obtain \( a_n = 3n - \frac{1}{n} \). This expression shows two critical components impacting the sequence: the linear term \( 3n \) and the diminishing fraction \( \frac{1}{n} \). The linear component grows larger as \( n \) increases, while the fraction becomes smaller. Thus, as \( n \) enlarges, the \( 3n \) term dominates, contributing to the idea of sequence increasing continually.
To comprehend the sequence's behavior, observe how each term is affected as we keep increasing \( n \). For the given sequence \( a_n = \frac{3n^2 - 1}{n} \), simplifying helps reveal its character more clearly.
On simplification, we obtain \( a_n = 3n - \frac{1}{n} \). This expression shows two critical components impacting the sequence: the linear term \( 3n \) and the diminishing fraction \( \frac{1}{n} \). The linear component grows larger as \( n \) increases, while the fraction becomes smaller. Thus, as \( n \) enlarges, the \( 3n \) term dominates, contributing to the idea of sequence increasing continually.
Infinite Limits
Infinite limits are crucial in discussing the behavior of sequences as they approach infinity. When we talk about "limits," we are predicting the value that a sequence approaches as \( n \) grows very large, tending towards infinity.
For the sequence \( a_n = \frac{3n^2 - 1}{n} \), simplifying this to \( a_n = 3n - \frac{1}{n} \), we can directly analyze its infinite limit. As \( n \rightarrow \infty \), the term \( \frac{1}{n} \) fades to zero because division between one and an infinitely large number becomes minuscule.
Consequently, our sequence simplifies to \( a_n \approx 3n \), suggesting that \( a_n \) will increase indefinitely, without settling at any fixed value. This realization highlights that the sequence limit is infinite, as its terms grow infinitely larger without bound.
For the sequence \( a_n = \frac{3n^2 - 1}{n} \), simplifying this to \( a_n = 3n - \frac{1}{n} \), we can directly analyze its infinite limit. As \( n \rightarrow \infty \), the term \( \frac{1}{n} \) fades to zero because division between one and an infinitely large number becomes minuscule.
Consequently, our sequence simplifies to \( a_n \approx 3n \), suggesting that \( a_n \) will increase indefinitely, without settling at any fixed value. This realization highlights that the sequence limit is infinite, as its terms grow infinitely larger without bound.
Sequence Analysis
Sequence analysis dives into understanding how sequences grow and behave. Breaking down the sequence \( a_n = 3n - \frac{1}{n} \) into its simplest form is the first step in comprehensive analysis.
The term \( 3n \) suggests a linear increase. Therefore, \( a_n \) appears to rise steadily as \( n \) escalates. To understand fully, always check smaller and larger \( n \) values, which ensures a complete grasp of how the sequence behaves throughout.
In our analysis, for very small \( n \), the effect of \( -\frac{1}{n} \) is more pronounced, potentially leading to negative values because this component is significant relative to \( 3n \). However, as \( n \) increases, \( 3n \) overshadow the diminishing \( -\frac{1}{n} \) influence, confirming that the sequence grows larger. This method of scrutinizing both extremes gives a complete picture of the sequence.
The term \( 3n \) suggests a linear increase. Therefore, \( a_n \) appears to rise steadily as \( n \) escalates. To understand fully, always check smaller and larger \( n \) values, which ensures a complete grasp of how the sequence behaves throughout.
In our analysis, for very small \( n \), the effect of \( -\frac{1}{n} \) is more pronounced, potentially leading to negative values because this component is significant relative to \( 3n \). However, as \( n \) increases, \( 3n \) overshadow the diminishing \( -\frac{1}{n} \) influence, confirming that the sequence grows larger. This method of scrutinizing both extremes gives a complete picture of the sequence.
Boundedness Evaluation
Boundedness evaluation determines whether a sequence stays within fixed bounds as it progresses. This requires checking if the sequence has limits both above and below.
In our sequence \( a_n = 3n - \frac{1}{n} \), examining as \( n \rightarrow \infty \) reveals that \( a_n \) is indeed not bounded above. Instead, it increases perpetually with no upper limit as demonstrated by \( 3n \) growing to infinity.
When considering if it’s bounded below, even for initial small values of \( n \), the sequence may yield negative outputs due to \( -\frac{1}{n} \). Hence, the sequence has no consistent lower bound. Thus, we assert that this sequence is not bounded above or below, establishing its unbounded nature in both directions.
In our sequence \( a_n = 3n - \frac{1}{n} \), examining as \( n \rightarrow \infty \) reveals that \( a_n \) is indeed not bounded above. Instead, it increases perpetually with no upper limit as demonstrated by \( 3n \) growing to infinity.
When considering if it’s bounded below, even for initial small values of \( n \), the sequence may yield negative outputs due to \( -\frac{1}{n} \). Hence, the sequence has no consistent lower bound. Thus, we assert that this sequence is not bounded above or below, establishing its unbounded nature in both directions.
