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Note that 14/33 = 0.4242424242 . This repeating decimal is obviously the same as the infinite sum 0.42 + 0.0042 + 0.000042 + 0.00000042 + 0.0000000042 + .
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OTHER SUMMATION FORMULAS Theorem A.7 Sum of the First n Positive Integers n n+1 + n = ------------------2 Note that the parameter n equals the number of terms in the sum. 1+2+3+ EXAMPLE A.8 Summing Positive Integers
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The sum of the first 10 integers is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 10(10+1)/2 = 55.
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[APPENDIX
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n n + 1 2n + 1 2 + n = ---------------------------------------6 The expression on the right appears to be a fraction. But it will always turn out to be an integer because it equals a sum of integers. 1 +2 +3 +
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Theorem A.8 Sum of the First n Squares
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EXAMPLE A.9 Summing Squares
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The sum of the first six squares is 12 + 22 + 32 + 42 + 52 + 62 = 6(7)(13)/6 = 546/6 = 91.
HARMONIC NUMBERS The harmonic series is the series of reciprocals:
1 -- = 1 + 1 + 1 + 1 + 1 + -- -- -- -- - - 2 3 4 5 k=1k
1 2 3 4 5 6 7 8 9 10
1.000000 1.500000 1.833333 2.083333 2.283333 2.450000 2.592857 2.717857 2.828968 2.928968
It is not hard to see that this series diverges. That is, its partial sums increase without bound. The partial sums of the harmonic series are called the harmonic numbers and are denoted by Hn :
1 -- = 1 + 1 + 1 + 1 + 1 + -- -- -- -- - - Hn = 2 3 4 5 k=1k
1 + -n
The first three harmonic numbers are
H1 = 1 -- = 1 k = 1k 1 -- = 1 + 1 = 3 --2 2 k = 1k
3 2 1
Table A.1 Harmonic numbers
H2 =
H3 =
1 -- = 1 + 1 + 1 = 5 -- -- -k 2 3 6
Although the harmonic numbers increase without bound, it is not obvious how fast they increase. Table A.1 suggests that they increase very slowly. The fact is that the harmonic numbers increase logarithmin n! cally: H n = (lgn). This means that they increase at about the 0 1 same rate as logarithmic numbers. More precisely, it means that 1 1 both ratios H n/ lgn and lgn /H n are bounded.
2 3 4 5 6 7 8 9 2 6 24 120 720 5040 40,320 362,880
STIRLING S FORMULA The factorial numbers frequently appear in the analysis of algorithms. They are defined by:
n! =
k = 1 2 3 4
The first ten factorials are shown in Table A.2.
Table A.2 Factorial numbers
APPENDIX]
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Unlike the harmonic sequence, the factorial sequence grows exponentially. This is reflected by Stirling s formula:
n! = 2n
n n -- e e
12n ,
where 0
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:
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