Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

The graph of $$ y=\frac{x ! e^{x}}{x^{x} \sqrt{2 \pi x}} $$ has a horizontal asymptote of \(y=1\). Use this fact to find an approximation for \(n !\) if \(n\) is a large positive integer.

Step-by-Step Solution

Answer

For large \(n\), use Stirling's approximation: \(n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n\).

Step 1: Identify the function components

The given function is \(y = \frac{x! e^{x}}{x^{x} \sqrt{2 \pi x}}\). It is composed of the factorial function \(x!\), the exponential function \(e^{x}\), a power \(x^{x}\), and the square root of a product \(\sqrt{2\pi x}\).

Step 2: Recall Stirling's Approximation

Stirling's approximation states that for large \(n\), \(n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n\). This approximation is based on the asymptotic behavior of \(n!\).

Step 3: Analyze the horizontal asymptote

The horizontal asymptote \(y=1\) implies that for large \(x\), the value of \(y\) approaches 1. Therefore, \(\frac{x! e^{x}}{x^{x} \sqrt{2 \pi x}} \approx 1\).

Step 4: Set up the equation

Given the horizontal asymptote, we have the equation \(x! e^{x} = x^{x} \sqrt{2\pi x}\). Rearrange it as \(x! = \frac{x^{x} \sqrt{2\pi x}}{e^{x}}\).

Step 5: Compare with Stirling's Approximation

Using Stirling's approximation \(n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n\), verify the equivalence \(\frac{n^{n} \sqrt{2\pi n}}{e^{n}} \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n\), confirming the expression matches this asymptotic form.

Key Concepts

Factorial Function

The factorial function, denoted as \(n!\), is an essential mathematical operation often used in permutations, combinations, and many areas of calculus and analysis. For a given positive integer \(n\):

• If \(n = 0\), then \(n! = 1\).
• For any positive integer \(n\), \(n! = n \times (n-1) \times (n-2) \times \ldots \times 1\).

In essence, the factorial function is the product of all positive integers less than or equal to \(n\). It rapidly grows larger as \(n\) increases, which is why it plays a significant role in topics such as Stirling's Approximation, helping us to approximate \(n!\) for large \(n\). This approximation becomes crucial because calculating the exact factorial for large \(n\) is computationally intensive.

Asymptotic Behavior

Asymptotic behavior refers to the way functions behave as their inputs approach a specific value, often infinity. In the context of Stirling's Approximation, it relates to how the factorial function \(n!\) behaves as \(n\) becomes large. Stirling's Approximation provides us with a way to express the factorial in terms of other simpler functions and constants:

• \(n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n\)

This indicates that for very large \(n\), the value of \(n!\) can be approximated accurately. This asymptotic expression allows mathematicians and scientists to bypass tedious calculations by utilizing an approximation that is much easier to compute, especially useful in fields like statistical mechanics or computer science.

Horizontal Asymptote

A horizontal asymptote of a function is a horizontal line \(y = c\) that the graph of the function approaches as \(x\) tends toward positive or negative infinity. For the function \(y = \frac{x! e^{x}}{x^{x} \sqrt{2 \pi x}}\), the horizontal asymptote is \(y = 1\). This means as \(x\) becomes very large, the function approaches the value 1. Understanding horizontal asymptotes is crucial because it helps us predict the behavior of complex functions without calculating them explicitly at infinity. In this particular example, knowing the behavior near this asymptote allows us to derive a more manageable form for \(n!\) using Stirling's Approximation. When analyzing functions, identifying the horizontal asymptote can thus give insights into the long-term trend of the function values.

Large Positive Integers

In mathematics, large positive integers typically refer to numbers that are significantly larger than typical everyday numbers. Handling such numbers often requires special approaches, like approximations, because direct computations become impractical. These integers are of particular interest when considering the factorial function, as \(n!\) grows extremely fast.Stirling's Approximation becomes relevant in this context, providing a feasible method to approximate the factorial values for large \(n\). Instead of performing numerous multiplications, Stirling's Approximation simplifies these calculations. Therefore, when solving problems involving large positive integers, relying on approximations and principles of asymptotic behavior can save time and computational resources. This is particularly beneficial in domains like algorithm analysis or probability theory, where understanding the behavior of large numbers is essential.