Question

The sequence \(\\{n !\\}\) ultimately grows faster than the sequence \(\left\\{b^{n}\right\\},\) for any \(b>1,\) as \(n \rightarrow \infty .\) However, \(b^{n}\) is generally greater than \(n !\) for small values of \(n\). Use a calculator to determine the smallest value of \(n\) such that \(n !>b^{n}\) for each of the cases \(b=2, b=e,\) and \(b=10\).

Short answer

Answer: For \(b=2\), the smallest value of n is 4; for \(b=e\), the smallest value of n is 4; and for \(b=10\), the smallest value of n is 8.

Step-by-step solution

Calculate Factorial and Exponential Terms

Calculate the terms for both sequences for increasing values of n, until \(n! > 2^n\). Start with n = 1 and then continue with n = 2, 3, 4, ... - When n = 1: \(1! = 1\) and \(2^1 = 2\) - When n = 2: \(2! = 2\) and \(2^2 = 4\) - When n = 3: \(3! = 6\) and \(2^3 = 8\) - When n = 4: \(4! = 24\) and \(2^4 = 16\) In this case, we find the smallest value of n = 4 where \(n! > 2^n\). #Case 2: b=e#

Calculate Factorial and Exponential Terms

Calculate the terms for both sequences for increasing values of n, until \(n! > e^n\). Use the value of e ≈ 2.71828. Start with n = 1 and then continue with n = 2, 3, 4, ... - When n = 1: \(1! = 1\) and \(e^1 = 2.71828\) - When n = 2: \(2! = 2\) and \(e^2 = 7.38906\) - When n = 3: \(3! = 6\) and \(e^3 = 20.08554\) - When n = 4: \(4! = 24\) and \(e^4 = 54.59815\) In this case, we find the smallest value of n = 4 where \(n! > e^n\). #Case 3: b=10#

Calculate Factorial and Exponential Terms

Calculate the terms for both sequences for increasing values of n, until \(n! > 10^n\). Start with n = 1 and then continue with n = 2, 3, 4, ... - When n = 1: \(1! = 1\) and \(10^1 = 10\) - When n = 2: \(2! = 2\) and \(10^2 = 100\) - When n = 3: \(3! = 6\) and \(10^3 = 1000\) - When n = 4: \(4! = 24\) and \(10^4 = 10000\) - When n = 5: \(5! = 120\) and \(10^5 = 100000\) - When n = 6: \(6! = 720\) and \(10^6 = 1000000\) - When n = 7: \(7! = 5040\) and \(10^7 = 10000000\) - When n = 8: \(8! = 40320\) and \(10^8 = 100000000\) In this case, we find the smallest value of n = 8 where \(n! > 10^n\). In conclusion, the smallest values of n for which the factorial sequence becomes larger than the exponential sequence in the given cases are: - For \(b=2\), n = 4 - For \(b=e\), n = 4 - For \(b=10\), n = 8

Key concepts

Exponential Growth

Exponential growth is a powerful mathematical concept where a quantity increases at a rate proportional to its current value. This type of growth is common in various real-world scenarios, like population growth or interest in banks. Essentially, exponential growth can seem slow initially, but it quickly becomes rapid and substantial as time goes on.
In mathematics, exponential growth is often represented by the expression \( b^n \), where \( b \) is the base, and \( n \) is the exponent. Here, \( b \) is greater than 1, which is crucial for the growth to be genuinely exponential. For instance, values like \( b=2 \), or \( b=e \) (Euler’s number approximately 2.718) are often used in problems dealing with exponential functions.
Exponential growth is integral to understanding sequence behavior, especially in comparison to other types of sequences like factorials. While factorials grow faster in the long run, exponential functions will generally exceed factorials at smaller values of \( n \). This aspect is important when tackling problems where you need to compare when a factorial exceeds an exponential sequence.

Sequence Comparison

In mathematics, comparing sequences involves examining how different sequences behave or grow as their variable, usually represented by \( n \), increases. For example, the factorial sequence \( n! \) and the exponential sequence \( b^n \) allow us to explore different growth behaviors.

Initially, for small \( n \), the exponential sequence \( b^n \) is typically larger because it accumulates rapidly due to its multiplicative nature. For example, with small numbers, \( 2^n \), \( e^n \), or \( 10^n \) can easily surpass \( n! \). However, as n increases, factorials, stacked products of numbers, grow significantly faster than exponentials, demonstrating a critical point about how sequences diverge.

In exercises comparing sequences, one might calculate individual terms, like \( n! \) and \( b^n \), to find the smallest \( n \) at which the factorial surpasses the exponential function. Despite initial appearances, the factorial sequence ultimately grows faster, illustrating an essential shift. Learning to compare these sequences can help with understanding various phenomena, including predicting when one process outpaces another.

Growth Rates

Growth rates describe how quickly a sequence or function increases as its input increases. Different mathematical expressions have characteristically different growth rates, impacting how they behave over time.
Factorials, denoted by \( n! \), showcase a particular kind of explosive growth because each term in the sequence multiplies by a new whole number. The result is that the growth rate of factorials is much faster than linear or even polynomial growth.
Comparing factorial growth with exponential growth is an interesting study. Though an exponential sequence like \( b^n \) is powerful, factorials eventually outpace it. This situation arises because a factorial grows by multiplying by each positive integer less than or equal to \( n \), rapidly becoming a larger sequence.

• For a given base \( b>1 \), exponential sequences have consistent multiplicative growth rates dictated by the constant base, contributing to their initial dominance.
• Meanwhile, factorials grow by adding more factors as \( n \) increases, allowing them to surpass even high exponential powers at larger \( n \). This shows the unique and potent growth rate of factorial sequences.

Understanding growth rates helps in various fields like economics, biology, and computer science, particularly in algorithms where determining efficiency is key. By mastering these concepts, students can better predict and explain sequence behaviors.