The factorial of a positive integer n can be computed as a pro...

Question

The factorial of a positive integer n can be computed as a product. n! = 1 * 2 * 3 *. . . * n

Calculators and computers can evaluate factorials very quickly. Before the days of modern technology, mathematicians developed Stirling’s formula for approximating large factorials. The formula involves the irrational numbers p and e.

n! = √2πn * n^n * e^−n

As an example, the exact value of 5! is 120, and Stirling’s formula gives the approximation as 118.019168 with a graphing calculator. This is “off” by less than 2, an error of only 1.65%. Work Exercises 59–62 in order. Use a calculator to find the exact value of 10! and its approximation, using Stirling’s

formula.

Accepted Answer

Welcome back. Everyone in this problem, we want to determine the exact value of eight factorial and write the approximation value using Sterling's formula. It says the exact value is 40,320. While the approximate value is 39,902 0.395 B says they are 322,560 39,902 0.395 respectively. C says they are 40,321 118,947,364 0.427. While D says they are 322,561 118,947,364 0.427 respectively. No, there are two things we want to find here. The exact value of eight factorial and the approximation value from Sterling's formula. Now, what do we know for both of those? Well, for a factorial recall, OK, that N factorial is going to be equal to N multiplied by N minus one, N minus two, N minus three, all the way down to three multiplied by two multiplied by one. So in this case, from that idea, then that means eight factorial is going to be equal to eight, multiplied by 765432 and one. OK. So we're finding the product of all of these numbers here. And when we go ahead and multiply we get eight factorial to be exactly equal to 40,320. Now how about the approximation value from Sterling's formula? Well, again, if you put that in blue recall that Sterling's formula tells us that the approximate value. OK. Approximate value let's say, well, let me write it here. Sterling's formula tells us that the approximate value is equal to two pi or the square root of two pi N multiplied by N to the power of N multiplied by E to the power of negative N. So in this case where N is it, that means by Sterling's formula is going to be equal to the square root of two pi multiplied by eight multiplied by eight to the power of it multiplied by E to the power of negative eight, which when we go ahead and multiply here, then we should get it to be approximately equal to 39,902 0.395. Thus the exact value for our factorial is 40,320. And the approximate value using Sterling's formula is 39,902 0.395. Therefore, A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Factorial

Stirling's Formula

Error Analysis

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