The factorial of a positive integer n can be computed as a product. n! = 1 * 2 * 3 *. . . * n
Calculators and computers can evaluate factorials very quickly. Before the days of modern technology, mathematicians developed Stirling’s formula for approximating large factorials. The formula involves the irrational numbers p and e.
n! = √2πn * n^n * 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%. Work Exercises 59–62 in order. Use a calculator to find the exact value of 10! and its approximation, using Stirling’s
formula.

Key Concepts

Factorial

The factorial of a positive integer n, denoted as n!, is the product of all positive integers from 1 to n. It is defined as n! = 1 × 2 × 3 × ... × n. Factorials are fundamental in combinatorics, probability, and various mathematical calculations, particularly in determining permutations and combinations.

Stirling's Formula

Stirling's formula provides an approximation for large factorials, expressed as n! ≈ √(2πn) * (n/e)^n. This formula is particularly useful when calculating the factorial of large numbers, as it simplifies the computation while maintaining a high degree of accuracy. It highlights the relationship between factorials and exponential functions.

Error Analysis

Error analysis in the context of approximations involves assessing the difference between the exact value and the estimated value provided by a formula like Stirling's. In the example given, the error is calculated as the absolute difference between 5! (120) and its approximation (118.019168), which is about 1.65%. Understanding error is crucial for evaluating the reliability of approximations in mathematical computations.