Question

Determine whether the sequence is bounded, bounded above, bounded below, or none of the above. $$\left\\{a_{n}\right\\}=\\{n \cos n\\}$$

Short answer

The sequence is neither bounded above nor below.

Step-by-step solution

Understanding the sequence

The given sequence is \( \{a_{n}\} = \{n \cos n\} \). This means each term of the sequence is the product of \( n \), where \( n \) is a natural number, and \( \cos n \), which is the cosine of the angle \( n \) (in radians).

Analyzing the range of cosine function

Recall that \( \cos n \) oscillates between \(-1\) and \(1\) for all values of \( n \). Therefore, \( n \cos n \) will oscillate between \(-n \) and \( n \).

Computing bounds

Since \( a_n = n \cos n \) is influenced by \( n \) and \( \cos n \), we see that the sequence will produce terms as large as \( n \) and as small as \(-n \) for each natural number \( n \). Thus, the sequence is not bounded from above because we can always find a larger value as \( n \) increases. Similarly, it is not bounded from below as \(-n\) becomes more negative for higher \( n \).

Concluding the bounds

Thus, considering both the increasing nature of \( n \) and the oscillating nature of \( \cos n \), the sequence \( \{ n \cos n \} \) does not have bounds; it is neither bounded above nor bounded below.

Key concepts

Sequence Analysis

Analyzing sequences is a fundamental aspect of mathematical studies. Put simply, a sequence is an ordered list of numbers, where each number is called a term. These terms can follow certain patterns or rules. Understanding such patterns is crucial for determining whether a sequence is bounded or not.

• A bounded sequence is limited; it does not exceed certain values.
• "Bounded above" means the sequence never goes beyond a certain largest number.
• "Bounded below" implies it never falls below a specific smallest number.

In our exercise, the sequence \( \{a_{n}\} = \{n \cos n\} \) is under investigation. Here, each term of the sequence, \(a_n\), is the result of multiplying a natural number \(n\) by the value of the cosine function. By examining these components, students can predict the sequence's behavior over time. In particular, due to the characteristics of \(n\) and its relation with the cosine function, this sequence lacks bounds, thus offering an insightful example of unbounded sequences.

Cosine Function

The cosine function is a fundamental trigonometric function that oscillates between the values \(-1\) and \(1\).* It plays a crucial role in sequence analysis when paired with natural numbers. Mathematically, the function is written as \(\cos(\theta)\), where \(\theta\) is the angle, often considered in radians for mathematical consistency.

• It is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) intervals.
• The oscillation of \(\cos n\) ensures that while \(n\) increases, producing either positive or negative product values depending on the cosine sign.
• This means that for \(\{n \cos n\}\), the amplitude of oscillation is expanded by the value of \(n\).

In the context of the sequence \(\{n \cos n\}\), the function impact is notable. As \(n\) grows, the oscillating outcomes of \(\cos n\) multiply, leading to potentially larger values. However, due to this inherent oscillation, even large declines occur, making the sequence unbounded.

Natural Numbers

The sequence \(\{a_{n}\} = \{n \cos n\}\) heavily relies on the properties of natural numbers. Natural numbers are the set of positive integers starting from 1: \(1, 2, 3,\) and so on. They are inherently infinite and increase without bound. This property is both their strength and a deciding factor in sequence analysis.

• When used in sequences, they provide an incrementally larger set of values.
• This unbounded growth displays no upper limit—as \(n\) progresses, so does its potential contribution to the sequence value.

By being paired with the cosine function, natural numbers influence the sequence by expanding both positive and negative outcomes. As a result, the unbounded nature of natural numbers prevents the sequence from being restricted, thus leading to an unbounded sequence. This interaction exemplifies how basic mathematical constructs can produce complex behaviors when combined in sequences.