Mathematics

Stirling’s Approximation: An Effective Technique for Estimating Factorials — Analysis

Background Information:

Factorial is a fundamental mathematical concept used to calculate the product of a given positive integer and all the positive integers below it. It is denoted by the symbol “!” and is commonly used in various mathematical and scientific applications. The factorial of a non-negative integer “n” is represented as “n!” and is defined as the product of all positive integers from 1 to “n”.
Symbolic representation: n! = n * (n — 1) * …………. * 2 * 1

But there is computational limitation while calculating factorial of a number using the above method if the number is large. To overcome this, there is Stirling’s formula or approximation. This powerful tool can calculate the value of factorial of large numbers. This approximation has a very invaluable impact in different fields not only in mathematics but also physics and computer science and more.

Symbolic Representation:

Stirling’s Approximation for n!

Proof of Stirling’s Approximation:

Here, we are going to prove Stirling’s approximation in a simple and easier way.

First of all, we start with the definition of factorial:

To obtain an approximation of this product, we can use the logarithm of both sides, taking advantage of the logarithm’s increasing nature.

Now, taking the fact into account that the sum of the logarithm is equal to the logarithm of the product:

Now, approximating the sum by integral, we get,

Now, integrating the above integral, we get,

Now, when we exponentiate both the sides, we get,

Hence, we here proved the Stirling’s approximation.

Accuracy of Stirling’s Approximation:

Comparing actual factorial with Stirling’s approximation (Source: Suraj Regmi’s Stirling’s Approximation Story)

Here, in the above graph, we see that as the value of n increases the gap between the the exact and approximate value goes on increasing. This might raise a question in our mind: How is it possible to claim that approximation goes on improving with increasing value of n? That’s a valid point!

Though the distance or absolute difference is increasing, the percentage of value difference is decreasing and the percentage of accuracy of the approximation is decreasing. The percentage of accuracy of the approximation is defined as:
|actual factorial — approximate factorial| / actual factorial

So, as n increases, the result has more percentage accuracy and close to the actual value. The more precise form of Stirling’s approximation is:

which is accurate to within 1% for N as small as 10 and becomes more accurate as N increases.

Conclusion:

In summary, Stirling’s approximation offers a highly accurate and powerful approach to approximate factorials of large numbers. Although it may not perform as well for small values of “n,” its accuracy significantly improves as “n” increases, making it well-suited for applications in fields like statistics, physics, and engineering. While alternative methods for factorial approximation exist, Stirling’s formula is widely embraced due to its simplicity and user-friendliness. It has made substantial contributions to mathematical theories and found practical utility across various domains. As a result, Stirling’s approximation is an invaluable tool relied upon by mathematicians, scientists, and engineers for effortlessly and reliably estimating factorials in a wide array of scenarios.

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