Ch. 7 - Further Topics in Algebra
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Chapter 8, Problem 59
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.
Verified step by step guidance1
Identify the exact value of 10! by calculating the product of all positive integers up to 10: 10! = 1 \(\times\) 2 \(\times\) 3 \(\times\) \(\ldots\) \(\times\) 10.
Use Stirling's approximation formula for factorials: n! \(\approx\) \(\sqrt{2\pi n}\) \(\times\) n^n \(\times\) e^{-n}.
Substitute n = 10 into Stirling's formula: 10! \(\approx\) \(\sqrt{2\pi \times 10}\) \(\times\) 10^{10} \(\times\) e^{-10}.
Calculate each component of the formula: \(\sqrt{2\pi \times 10}\), 10^{10}, and e^{-10}.
Multiply the results from the previous step to find the approximate value of 10! using Stirling's formula.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factorial
The factorial of a positive integer n, denoted as n!, is the product of all positive integers from 1 to n. It is defined as n! = 1 × 2 × 3 × ... × n. Factorials are fundamental in combinatorics, probability, and various mathematical calculations, particularly in determining permutations and combinations.
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Factorials
Stirling's Formula
Stirling's formula provides an approximation for large factorials, expressed as n! ≈ √(2πn) * (n/e)^n. This formula is particularly useful when calculating the factorial of large numbers, as it simplifies the computation while maintaining a high degree of accuracy. It highlights the relationship between factorials and exponential functions.
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Error Analysis
Error analysis in the context of approximations involves assessing the difference between the exact value and the estimated value provided by a formula like Stirling's. In the example given, the error is calculated as the absolute difference between 5! (120) and its approximation (118.019168), which is about 1.65%. Understanding error is crucial for evaluating the reliability of approximations in mathematical computations.
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