Problem 109 - Calculus Early Transcendentals

Question

In the following exercises, two sequences are given, one of which initially has smaller values, but eventually "overtakes" the other sequence. Find the sequence with the larger growth rate and the value of \(n\) at which it overtakes the other sequence. $$a_{n}=n ! \text { and } b_{n}=n^{0.7 n}, n \geq 2$$

Step-by-Step Solution

Answer

Answer: The sequence \(a_n = n!\) has a larger growth rate initially, and the \(b_n = n^{0.7n}\) sequence overtakes it at \(n = 13\).

Step 1: Observe the Behavior of the Sequences

Our first step is to find out how \(a_n\) and \(b_n\) grow as \(n\) increases. Let's see the behavior of both the sequences: $$a_n = n! = 2 \cdot 3 \cdot 4 \cdots n$$ $$b_n = n^{0.7n}$$ a) As n increases, the factorial \(n!\) gets multiplied by increasing integers, so the sequence \(a_n\) grows very quickly. b) The sequence \(b_n = n^{0.7n}\) also grows with increasing \(n\), but at a slower rate than the factorial sequence. Therefore, initially, the sequence \(b_n\) has smaller values than \(a_n\).

Step 2: Find the Intersection Point

Now, we need to find the value of \(n\) at which the \(b_n\) sequence overtakes the \(a_n\) sequence. We'll set up an inequality and solve for \(n\): $$b_n = n^{0.7n} > a_n = n!$$ To make this inequality easier to solve, let's take the natural logarithm of both sides: $$0.7n \ln n > \ln(n!)$$ We'll use Stirling's approximation for the natural logarithm of a factorial, which states that \(\ln(n!) \approx n \ln n - n\): $$0.7n \ln n > n \ln n - n$$ Now let's solve for \(n\). Rearrange the inequality: $$0.3n \ln n > n$$ Divide both sides by \(0.3n\): $$\ln n > \dfrac{1}{0.3} \Rightarrow \ln n > \dfrac{10}{3}$$ As the natural logarithm function is increasing, we just need to find the smallest integer \(n\) such that \(\ln n > \dfrac{10}{3}\): Taking the exponential of both sides: $$n > e^{\frac{10}{3}} \approx 12.13$$ The smallest integer greater than \(12.13\) is \(n = 13\). So the value of \(n\) at which the \(b_n\) sequence overtakes the \(a_n\) sequence is \(13\).

Key Concepts

Convergence and Divergence in Sequences

In calculus, sequences are lists of numbers generated by some mathematical rule and can either approach a specific value or grow indefinitely as the number of terms increases. This behavior is referred to as convergence and divergence, respectively. A sequence converges if, as the number of terms goes to infinity, the terms approach a finite limit. In contrast, a sequence diverges if it does not approach a specific value or if it increases without bound.

For example, when comparing sequences like factorials and exponentially growing functions, determining which one grows faster can help us understand where one diverges more aggressively than the other. This type of analysis is important when analyzing algorithms in computer science or growth processes in biology, where one must discern sustainable growth patterns from those that are unsustainable.

Factorial Growth

The factorial of a number, denoted by an exclamation point (!), is the product of all positive integers from 1 up to that number and is a fundamental concept in mathematics, particularly in permutations and combinations. Factorial growth is dramatic and outpaces most other algebraic growth rates. For instance, the value of a factorial sequence such as 5! is calculated as 5 x 4 x 3 x 2 x 1 = 120. This type of growth becomes astronomical as the value of n increases.

When analyzing sequences with factorials, it's clear that even for small values of n, the numbers quickly become very large, which is illustrative of faculty growth's explosively divergent nature. It's a clear-cut example of a sequence that often overtakes others with different growth rates as n becomes large.

Exponential Growth

In contrast to factorial growth, exponential growth occurs when a quantity increases by the same proportion (the base of the exponent) in each unit of time. For example, a sequence represented by ^n is said to grow exponentially. This type of growth is pervasive throughout both natural processes, like population growth, and artificial systems, like compound interest. While exponential growth is rapid, it doesn't usually increase as swiftly as factorial growth.

Regarding the provided exercise, the term ^{0.7n} represents a sequence with exponential growth but with a slower rate than the factorial sequence. This slower start often misleads when trying to understand the comparative long-term behavior of different sequences.

Stirling's Approximation

Stirling's approximation is a powerful tool in mathematics for estimating the factorial of a large number. It approximates the factorial function by using a formula that involves natural logarithms and the value of n. Mathematically, it's often expressed as:

\[ ! \right) \]

This approximation becomes incredibly useful when dealing with large numbers because calculating factorials directly can quickly become impractical. In the context of the solution provided in the exercise, Stirling's approximation has been utilised to simplify the comparison between the factorial growth and the exponentially growing sequence to determine the point of overtaking.