Question

Determine whether the sequence is bounded, bounded above, bounded below, or none of the above. $$\left\\{a_{n}\right\\}=\left\\{2^{n}-n !\right\\}$$

Short answer

The sequence is not bounded above or below.

Step-by-step solution

Understand the Sequence

The sequence is given by \( \{a_n\} = \{2^n - n!\} \). It consists of the terms resulting from subtracting the factorial of \( n \) from \( 2^n \) for each natural number \( n \).

Analyze the Initial Terms

Calculate the first few terms: \( a_1 = 2^1 - 1! = 2 - 1 = 1 \), \( a_2 = 2^2 - 2! = 4 - 2 = 2 \), \( a_3 = 2^3 - 3! = 8 - 6 = 2 \), and \( a_4 = 2^4 - 4! = 16 - 24 = -8 \), ...Notice that as \( n \) increases, the factorial grows faster than the exponential term initially.

Determine the Growth Rates

For large \( n \), compare \( 2^n \) and \( n! \). The factorial \( n! \) grows faster than the exponential \( 2^n \) because factorial involves multiplying through all integers up to \( n \), whereas \( 2^n \) doubles a fixed number \( n \) times. Therefore, \( 2^n - n! \) will eventually become negative and decrease without bound.

Conclusion about Bounds

Since for large \( n \), the sequence terms become increasingly negative, \( \{a_n\} = \{2^n - n!\} \) is not bounded below, as it can reach arbitrarily large negative values. Initially, the sequence has positive terms, indicating it is not consistently going to infinity or remaining above a certain negative value. This means the sequence \( \{a_n\} \) is not bounded above, bounded below, or bounded because it eventually decreases without limit.

Key concepts

Sequence Analysis

Sequences are fundamental in mathematics, acting as ordered lists of numbers often denoted as \( \{a_n\} \). These sequences have rules or functions that define their terms in relation to their position within the sequence. In the case of the sequence \( \{a_n\} = \{2^n - n!\} \), each term is calculated by taking \( 2^n \) and subtracting \( n! \).

Sequence analysis is the process of understanding the behavior and properties of these terms as \( n \) increases. By analyzing sequences, mathematicians can determine various features, such as whether sequences converge to a limit, whether they are finite or infinite, and whether they are bounded.

In our given sequence, we start by calculating the first few terms to observe their pattern:

• For \( n = 1 \), \( a_1 = 2^1 - 1! = 1 \)
• For \( n = 2 \), \( a_2 = 2^2 - 2! = 2 \)
• For \( n = 3 \), \( a_3 = 2^3 - 3! = 2 \)
• For \( n = 4 \), \( a_4 = 2^4 - 4! = -8 \)

As seen, initially, the terms are positive, but at \( n = 4 \), they become negative, indicating a shift in behavior.

By exploring further terms, one can predict the growth behavior and determine whether a sequence has bounds or limits.

Factorial Growth

Factorial growth is a type of mathematical growth where the function involves a factorial term \( n! \). A factorial, which is a product of all positive integers less than or equal to \( n \), grows extremely fast.

For instance, in our sequence \( 2^n - n! \), the factorial component \( n! \) quickly outpaces \( 2^n \). Initially, for small \( n \), \( 2^n \) might be larger, but factorial growth soon surpasses it by leaps and bounds.

• At \( n = 3 \), \( 3! = 6 \), while \( 2^3 = 8 \)
• At \( n = 4 \), \( 4! = 24 \), while \( 2^4 = 16, \) resulting in a negative value.

The rapid acceleration of factorial growth leads to the overtaking of the exponential component \( 2^n \) in a sequence. This results in parts of the sequence becoming significantly negative as \( n \) increases.

Understanding the nature of factorial growth is vital in determining sequence behavior, particularly when comparing growth rates with exponential or linear functions.

Exponential Growth

Exponential growth refers to the situation where the value of a function increases at a rate proportional to its current value. In mathematical terms, it is often represented as \( b^n \), where \( b \) is a base greater than 1, like 2 in our sequence.

In the sequence \( \{2^n - n!\} \), \( 2^n \) represents the exponential growth portion. It involves repeated multiplication by 2 as \( n \) increases.

• For \( n = 1 \), \( 2^1 = 2 \)
• For \( n = 3 \), \( 2^3 = 8 \)
• For \( n = 5 \), \( 2^5 = 32 \)

Initially, exponential functions like \( 2^n \) can grow rapidly. However, when compared with factorials, the exponential growth may not be as significant as factorial growth for larger \( n \). This comparison is crucial in our sequence to understand why it eventually becomes negative.

Exponential growth is fundamental to many fields, including finance, population studies, and natural sciences. It helps in predicting behaviors and outcomes where growth rate is continually compounded.