Use a pocket calculator to check the accuracy of Stirling's approximation for N=50 . Also check

Use a pocket calculator to check the accuracy of Stirling's...

Key Concepts

Stirling's Approximation

Stirling's approximation is a formula used to approximate large factorials with a simpler, continuous function. It is an essential tool in mathematics, statistics, and physics, as it simplifies the computation and analysis of combinatorial expressions and probability distributions when dealing with large numbers. This approximation helps bridge the gap between discrete and continuous analysis, offering insight into the asymptotic behavior of factorial-based functions.

Factorial Function

The factorial function, defined for any non-negative integer N as the product of all positive integers up to N, grows very quickly and becomes computationally challenging for large values. Understanding the properties of the factorial function is crucial in combinatorics, probability, and many areas of mathematical analysis, where approximations like Stirling's formula are employed to simplify complex calculations.

Logarithmic Transformation of Factorials

Taking the logarithm of the factorial, ln N!, converts multiplicative relationships into additive ones, which is particularly useful when dealing with the large numbers produced by the factorial function. This logarithmic transformation often makes it easier to analyze and approximate the behavior of factorials, as seen in the application of Stirling's approximation and related equations.

Error Analysis in Asymptotic Approximations

Error analysis involves assessing the difference between an approximation and the exact value of a quantity. In the context of Stirling's approximation and logarithmic approximations like the one given in equation 2.16, error analysis provides insight into the accuracy and limitations of these approximations, especially for specific values of N. This understanding is key in many applied mathematical fields where precise calculations are critical.

Transcript

00:01 And this problem, we're going to go over problem 2 .15 in schroeder's thermal physics textbook.

00:09 So we're going to check the accuracy of two approximations.

00:14 Well, first is sterling's approximation of the permutation of some number, which we're going to call n.

00:31 And this number is much, much greater than one.

00:36 So the permutation of n in the sterling's approximation will be approximately equal to n to the power of n times e to the power of negative n times square root of 2 pi n.

01:05 And in this problem, we're going to check it for n equals 50.

01:15 So we're going to compare by using a calculator what the permutation of 50 is with its approximation on the right hand side.

01:33 So we're going to calculate this, and we're also going to calculate this.

01:37 For n is equal to 50.

01:53 So typing this into the calculator, if you don't remember how to type in a permutation, the explanation point, just go to math, click on math, and then go to the right to click on probability and then hit 4 where you will see the explanation point if you're using a ti graphing calculator so what we get for the permutation of 50 is about 3 .04 .141 times 10 to the 64 and we will get for the approximation of this which is sterling's approximation is 3 .03634 times 10 to the 64.

02:51 And we also want to find, in order to check the accuracy of sterling's approximation, we're going to find the percent error.

03:04 So that will be the difference between these two calculations divided by the permutation of 50.

03:28 So we go, we can write the permutation of 50 minus its approximation, well, where we will just plug in the calculate, what we've calculated them to be, divided by the permutation of 50.

04:01 And what you should get, you should get that there is a 0 .1665 % error.

04:13 For sterling's approximation, using where n equals 50...

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