Theory and ExamplesThe n th root of n !a. Show that and

Chapter 10, Problem 104E

(choose chapter or problem)

Problem 104E

Theory and Examples

The n th root of n !

a. Show that and hence, using Stirling’s approximation (Chapter 8, Additional Exercise 32a), that

b. Test the approximation in part (a) for n = 40, 50, 60,…., as far as your calculator will allow.

Reference: Chapter 8, Additional Exercise 32a

The Gamma Function and Stirling’s Formula.

Stirling’s formula Scottish mathematician James Stirling (1692–1770) showed that

so, for large x,

Dropping  leads to the approximation

a. Stirling’s approximation for n ! Use Equation (3) and the fact that to show that

As you will see if you do Exercise 104 in Section 10.1, Equation (4) leads to the approximation(5)

b. Compare your calculator’s value for n! with the value given by Stirling’s approximation for n = 10, 20, 30,…., as far as your calculator can go.

c. A refinement of Equation (2) gives

Compare the values given for 10! by your calculator, Stirling’s approximation, and Equation (6).