Enter the number and the base of the natural logarithm into the calculator to determine Stirling’s Approximation.

Stirling’s Approximation Formula

The following formula is used to calculate Stirling’s Approximation.

S(n) = sqrt(2πn) * (n/e)^n


  • S(n) is the Stirling’s Approximation of n n is the number for which we are calculating the approximation e is the base of the natural logarithm (approximately equal to 2.71828) π is a mathematical constant (approximately equal to 3.14159)

To calculate Stirling’s Approximation, multiply 2 by π and then take the square root of the product. Multiply this result by n raised to the power of n divided by e. The final result is Stirling’s Approximation of n.

What is a Stirling’s Approximation?

Stirling’s Approximation is a mathematical formula used to approximate the factorial of a large number. It is named after the Scottish mathematician James Stirling. The approximation states that the factorial of a number n is approximately equal to the square root of 2πn times (n/e) to the power of n. This approximation becomes more accurate as n increases, and is particularly useful in fields such as statistics and combinatorics where factorials of large numbers are often encountered.

How to Calculate Stirling’S Approximation?

The following steps outline how to calculate Stirling’s Approximation:

  1. First, determine the value of n.
  2. Next, calculate n/e.
  3. Next, raise n/e to the power of n.
  4. Next, calculate sqrt(2πn).
  5. Finally, multiply sqrt(2πn) by (n/e)^n to get the value of S(n).

Example Problem:

Use the following variables as an example problem to test your knowledge:

n = 5

e = 2.71828

π = 3.14159