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Unformatted text preview: 3.1. PERMUTATIONS 81 n n ! Approximation Ratio 1 1 .922 1.084 2 2 1.919 1.042 3 6 5.836 1.028 4 24 23.506 1.021 5 120 118.019 1.016 6 720 710.078 1.013 7 5040 4980.396 1.011 8 40320 39902.395 1.010 9 362880 359536.873 1.009 10 3628800 3598696.619 1.008 Table 3.4: Stirling approximations to the factorial function. Definition 3.3 Let a n and b n be two sequences of numbers. We say that a n is asymptotically equal to b n , and write a n ∼ b n , if lim n →∞ a n b n = 1 . Example 3.4 If a n = n + √ n and b n = n then, since a n /b n = 1 + 1 / √ n and this ratio tends to 1 as n tends to infinity, we have a n ∼ b n . Theorem 3.3 (Stirling’s Formula) The sequence n ! is asymptotically equal to n n e n √ 2 πn . The proof of Stirling’s formula may be found in most analysis texts. Let us verify this approximation by using the computer. The program StirlingApprox imations prints n !, the Stirling approximation, and, finally, the ratio of these two numbers. Sample output of this program is shown in Table 3.4. Note that, while the ratio of the numbers is getting closer to 1, the difference between the exact...
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 Spring '09
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