Problem 49 - A Graphical Approach to Precalculus with Limits

Question

The factorial of a positive integer \(n\) can be computed as a product: \(n !=1 \cdot 2 \cdot 3 \cdot \cdots \cdot n\) Calculators and computers can evaluate factorials quickly. Before the days of techmology. mathematicians developed a formula, called Stirling's formula, for approximating large factorials. Interestingly enough, it imolves the irrational numbers \(\pi\) and \(e\). $$n ! \approx \sqrt{2 \pi n} \cdot n^{n} \cdot e^{-n}$$ As an example, the exact value of \(5 !\) is \(120,\) and Stirling's formula gives the approximation as 118.019168 with a graphing calculator. This is " off" by less than \(2,\) an error of only \(1.65 \%\) Use a calculator and Stirling's formula to find the exact value of \(10 !\) and its approximation.

Step-by-Step Solution

Answer

The exact 10! is 362880; Stirling's approximation is 359869.7.

Step 1: Compute Exact Value of 10!

The factorial of 10, denoted as \(10!\), is calculated by multiplying all integers from 1 to 10. The computation is as follows:\[10! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10 = 362880\]

Step 2: Apply Stirling's Approximation Formula

Use Stirling's approximation formula to estimate \(10!\):\[\]\[10! \approx \sqrt{2 \pi \times 10} \cdot 10^{10} \cdot e^{-10}\]Firstly, compute \(\sqrt{2 \pi \times 10}\):\[\sqrt{2 \pi \times 10} = 7.9788451\]Then, compute \(10^{10}\):\[10^{10} = 10000000000\]Next, compute \(e^{-10}\):\[e^{-10} \approx 0.00004539993\]Finally, multiply the results together to get the approximation:\[7.9788451 \times 10000000000 \times 0.00004539993 \approx 359869.7169\]

Step 3: Compare Results

Now, compare the exact value of \(10!\) which is 362880 with the Stirling's approximation which is 359869.7169.The approximation is quite close to the exact value but slightly less in this case.

Key Concepts

Factorials

When considering mathematical operations, the factorial is a simple yet powerful concept. Factorials are symbolized by an exclamation mark "!" and describe the product of all positive integers up to a specific number. For instance, the factorial of 5, represented as 5!, is calculated as follows:

• 5! = 1 × 2 × 3 × 4 × 5 = 120

This process involves straightforward multiplication, making it a fundamental element in combinatorics and probability theory. Beyond its direct calculations, factorials play a significant role in calculating permutations and combinations. As numbers increase, factorials grow rapidly—making precise computation without the aid of calculators quite challenging. Thankfully, mathematical approximations, like Stirling's, offer practical solutions for large numbers.

Mathematical Approximation

Mathematical approximation comes into play when exact calculations are unfeasible or unnecessary. One famous approximation, Stirling's Approximation, provides a practical method to estimate large factorials. This approximation is particularly helpful since factorials grow exponentially with higher numbers. Stirling's Approximation leverages irrational numbers to achieve a remarkable level of accuracy by using the following formula:

• \[n ! \approx \sqrt{2 \pi n} \cdot n^{n} \cdot e^{-n}\]

The application of this formula involves calculating each component precisely, such as the square root term, powers, and exponentials. Despite being an approximation, it achieves strikingly close values to the actual numbers, as seen from comparing the Stirling approximation of 5! to its exact value. The formula beautifully integrates both mathematical elegance and practical application.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction. They play a crucial role in many mathematical formulas, including Stirling's Approximation. Two notable irrational numbers in mathematics are \(\pi\) and \(e\).

• \(\pi\): This number represents the ratio of a circle's circumference to its diameter and is approximately equal to 3.14159.


• \(e\): Known as Euler's number, it is about 2.71828 and is vital in the fields of calculus and exponential growth.

Their presence in Stirling's formula highlights their utility beyond theoretical constructs. These numbers help stabilize the approximation and contribute to its accuracy. Understanding their properties helps demystify how a formula can estimate such complex calculations with impressive precision.