vb.net print barcode zebra The first ten factorials are shown in Table A.2. in Java

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The first ten factorials are shown in Table A.2.
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Table A.2 Factorial numbers
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APPENDIX]
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ESSENTIAL MATHEMATICS
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Unlike the harmonic sequence, the factorial sequence grows exponentially. This is reflected by Stirling s formula:
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n! = 2n
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The value of the variable depends upon n, but in any case it is bounded between 0 and 1. Thus, for large n, the exponent /12n will be very close to 0, making the factor e /12n very close to 1. Consequently, Stirling s formula is often expressed in this simpler approximate form: n n n! 2n -e The factorial numbers grow exponentially: n! = (2n). This fact follows from Stirling s formula. Another important consequence of Stirling s formula is that lg(n!) is asymptotically equivalent to n lgn: lg(n!) = (n lgn). FIBONACCI NUMBERS The Fibonacci numbers also frequently appear in the analysis of algorithms. They are defined by:
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Fn = 0, if n = 0 1, if n = 1 F n 1 + F n 2 , if n > 1
0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 1 2 3 5 8 13 21 34 55 89 144
The first 13 Fibonacci numbers are shown in Table A.3. Like the factorial sequence, the Fibonacci sequence grows exponentially, as is verified by De Moivre s formula:
F n = ----------------- , where 5
1+ 5 = --------------- , and 2
1 5 = --------------2
Thus, Fn = ( ). Here, = 1.618034 and = 0.618034. These two constants are the golden mean and its conjugate.
Table A.3 Fibonacci numbers
Review Questions
A.1 A.2 A.3 A.4 A.5 A.6 A function f() is called idempotent if f(f(x)) = f(x) for all x in the domain of f(). Explain why the floor and ceiling functions are idempotent. What is a logarithm What is the difference between weak induction and strong induction How can you decide when to use strong induction What is Euler s constant What makes Stirling s formula useful
ESSENTIAL MATHEMATICS
[APPENDIX
Problems
A.1 A.2 A.3 Prove Theorem A.1 on page 319. Prove Theorem A.2 on page 320. True or false: a. f = o(g) g = (f) b. f = O(g) g = (f) g = (f) c. f = (g) d. f = O(g) f = (g) f = (g) e. f = (g) f. f = (h) g = (h) g. f = (h) g = (h) h. n 2 O(n lg n) i. n 2 (n lg n) j. n 2 (n lg n) k. lg n (n) l. lg n o(n)
f + g = (h) f g = (h)
A.4 A.5 A.6 A.7
Prove Theorem A.5 on page 323. Prove Theorem A.6 on page 323. Prove Theorem A.7 on page 323. Run a program that tests De Moivre s formula on page 325 by comparing the values obtained from it with those obtained from the recursive definition of the Fibonacci numbers.
Answers to Review Questions
A.1 A.2 A.3 The floor and ceiling functions are idempotent because they return integer values, and according to Theorem A.1 on page 319, the floor or ceiling of an integer is itself. A logarithm is an exponent. It is the exponent on the given base that produces the given value. The First Principle of Mathematical Induction ( weak induction) allows the inductive hypothesis that assumes that the proposition P(n) is true for some single value of n. The Second Principle of Mathematical Induction ( strong induction) allows the inductive hypothesis that assumes that all the propositions P(k) are true for all k less than or equal to some value of n. Use weak induction (the first principle) when the proposition P(n) can be directly related to its predecessor P(n 1). Use strong induction (the second principle) when the proposition P(n) depends upon P(k) for k < n 1. Euler s constant is the limit of the difference (1 + 1/2 + 1/3 + . . . + 1/n) lnn. Its value is approximately 0.5772. Stirling s formula is a useful method for approximating n! for large n (e.g., n > 20).
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