Formulas involving n! often arise in applications. However, these formulas are usually inconvenient to work with because there are difficult to simplify algebraically. In this project, you will study a formula used to approximate n!. The formula is named after the Scottish mathematician James Stirling, though it may have been known earlier to the English mathematician de Moivre.

1. Notice that n! grows incredibly fast. Complete the following table.
2. Show that n! grows so quickly that (n+1)! grows faster than n!. That is, show that

   lim n ® ¥  (n+1)! n! = ¥.

   Compare this behavior to that of (n+1)² and n², (n+1)³ and n³, (n+1)⁴ and n⁴, etc.
3. Determine the Maclaurin series for the function f(x) = ln(1+x). Show that ln(1+x) is equal to its Maclaurin series on the interval (-1,1].
4. Suppose p is a positive number.
   1. Use your result from Problem 3 to determine the Maclaurin series for
   2. Find the interval of convergence of the series.
   3. Argue that (We are especially interested in large values of p.)
5. One can show that p! = ò₀^¥ x^pe^-x  dx for any nonnegative integer p. Use the tabular method of integration by parts to verify this for p = 4.
6. Show that after writing x^p = e^plnx and substituting x = p+yÖp, we get
7. Use Problem 4c to find an approximation for the integral on the right hand side.
8. By cleverly using multiple integrals (as you'll do in Calculus III), one can show that Assume this and use it, along with your result from Problem 7, to show that
9. Show that
10. You have successfully derived Stirling's formula: Complete the following table. Notice that the approximation gets better as n gets larger. In fact,
11. Use Stirling's formula in each of the following problems.
    1. Show that lim_{n ® ¥} [ⁿÖn!] = ¥.
    2. Find the limit of the sequence {[n!/( nⁿ)] }.
    3. Use the root test to determine whether the series å_{n = 0}^¥  [(eⁿ)/ n!] converges or diverges.
12. In statistical mechanics, the approximation lnN! » N lnN - N is often used, where N is on the order of Avogadro's number. Use Stirling's formula to approximate lnN! and thereby justify the above approximation.