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

Variables:

- 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:

- First, determine the value of n.
- Next, calculate n/e.
- Next, raise n/e to the power of n.
- Next, calculate sqrt(2πn).
- 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