The factorial of a positive integer n can be computed as a product.

n! = 1 · 2 · 3 · g · 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 · nⁿ · 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%.

Repeat Exercises 59 and 60 for n = 13. What seems to happen as n gets larger?

Exercises 59.

Use a calculator to find the exact value of 10! and its approximation, using Stirling’s formula.

Exercises 60.

Subtract the smaller value from the larger value in Exercise 59. Divide it by 10! and convert to a percent. What is the percent error to three decimal places?