# How to proof stirling approximation

• MHB
• Another1
In summary, Stirling's approximation is a mathematical formula that approximates the value of the factorial of a number. It is based on the idea that the logarithm of the factorial is approximately equal to the integral of the natural logarithm of x from 1 to n. The wiki page provides formal proofs and error bounds for this approximation. It should be noted that the statement "ln(x!) = xln(x) - x" is not true, but rather an approximation.
Another1
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i want to know about stirling approximation. why $$\displaystyle lnx! = xlnx - x$$

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Another said:
i want to know about stirling approximation. why $$\displaystyle lnx! = xlnx - x$$

Hi Another!

Wiki explains Stirling's approximation.

We can see that it's true because:
$$\ln n! = \sum_{k=1}^n \ln k \approx \int_1^n \ln x\,dx = (x\ln x - x)\Big|_1^n = n\ln n - n + 1$$
The wiki page has formal proofs and bounds on the error.

First, do you understand that "Stirling's Approximation" is an approximation. There is NO proof that "ln(x!)= xln(x)- x" because that is NOT true- they are approximately equal, not equal.

## 1. What is the Stirling approximation?

The Stirling approximation is a mathematical formula used to approximate the factorial of a large number. It was first derived by Scottish mathematician James Stirling in the 18th century.

## 2. How is the Stirling approximation derived?

The Stirling approximation is derived using the Euler-Maclaurin formula, which is a method for approximating definite integrals. It involves taking the logarithm of the factorial and then using a series expansion to simplify the expression.

## 3. When is the Stirling approximation most accurate?

The Stirling approximation is most accurate when the number being approximated is large. As the number gets larger, the relative error of the approximation decreases. However, for smaller numbers, the approximation may not be as accurate.

## 4. What are the limitations of the Stirling approximation?

The Stirling approximation is only accurate for large numbers and may not be as accurate for smaller numbers. Additionally, it is an approximation and not an exact solution, so there will always be some margin of error. It is also not suitable for complex or imaginary numbers.

## 5. How is the Stirling approximation used in scientific research?

The Stirling approximation is commonly used in scientific research, particularly in fields such as physics and statistics. It is used to simplify complex mathematical expressions and make calculations easier. It is also used in the analysis of algorithms and in the study of asymptotic behavior.

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