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ESSENTIAL MATHEMATICS
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GEOMETRIC SERIES A series is a sequence of possibly infinitely many terms whose sum is to be determined. A geometric series is a series in which each term is the same multiple of its predecessor. For example, 20 + 60 + 180 + 540 + 1620 + 4860 + is a geometric series because each term is 3 times the size of its predecessor. The multiplier 3 is called the common ratio of the series. Theorem A.5 Sum of a Finite Geometric Series If r 1, then a 1 rn = --------------------1 r Here, a is the first term in the series, r is the common ratio, and n is the number of terms in the series. a + ar + ar + ar +
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+ ar
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EXAMPLE A.6 Finite Geometric Series
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For the sum 20 + 60 + 180 + 540 + 1620 + 4860, the three parameters are a = 20, r = 3, and n = 6. So the sum is a 1 r 20 1 3 20 1 729 20 729 -------------------- = ------------------------ = ---------------------------- = ---------------------- = 7280 1 r 1 3 2 2
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Theorem A.6 Sum of an Infinite Geometric Series If 1 < r < 1, then a 2 3 a + ar + ar + ar + = ---------1 r EXAMPLE A.7 Infinite Geometric Series
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For the sum 0.42 + 0.0042 + 0.000042 + 0.00000042 + 0.0000000042 + a = 0.42 and r = 0.01. So the infinite sum is a0.42 ---------- = ------------------ = 0.42 = 42 = 14 ----------------1 r 1 0.01 0.99 99 33 , the three parameters are
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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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ESSENTIAL MATHEMATICS
[APPENDIX
Theorem A.8 Sum of the First n Squares 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 +
2 2 2
EXAMPLE A.9 Summing Squares
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 1 -- = 1 + 1 + 1 = 5 -- -- -2 3 6 k = 1k
3 2 1
Table A.1 Harmonic numbers
H2 =
H3 =
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
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