ics141-lecture10-Summations

# ics141-lecture10-Summations - University of Hawaii ICS141...

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10-1 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii ICS141: Discrete Mathematics for Computer Science I Department of Information and Computer Sciences University of Hawaii Stephen Y. Itoga

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10-2 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Lecture 10 Chapter 2. Basic Structures 2.4 Sequences and Summations
10-3 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Some material in these slides were taken/adapted from the slides made by Prof. Michael P. Frank and Prof. Jonathan L. Gross which are provided through the publisher of “Discrete Mathematics and Its Applications” written by Kenneth H. Rosen. Some material is from Prof. Baek

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10-4 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Summation Notation Given a sequence { a n } , an integer lower bound (or limit ) j 0 , and an integer upper bound k j , then the summation of { a n } from a j to a k is written and defined as follows: k j j k j i i a a a a + + + = + = ... 1 = = = = = k j l l k j m m k j i i a a a
10-5 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii Generalized Summations For an infinite sequence, we may write: To sum a function over all members of a set X ={ x 1 , x 2 , …} : Or, if X ={ x | P ( x )}, we may just write: + + = ) ( ) ( ) ( 2 1 x f x f x f X x + + = ) ( ) ( ) ( 2 1 ) ( x f x f x f x P + + = + = 1 j j j i i a a a

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10-6 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii ( 29 32 17 10 5 ) 1 16 ( ) 1 9 ( ) 1 4 ( ) 1 4 ( ) 1 3 ( ) 1 2 ( 1 2 2 2 4 2 2 = + + = + + + + + = + + + + + = + = i i Simple Summation Example = 100 1 1 i i = + + + + 100 1 4 1 3 1 2 1 1
10-7 ICS 141: Discrete Mathematics I (Spr 2008) University of Hawaii More Summation Examples An infinite sequence with a finite sum: Using a predicate to define a set of elements to sum over: 87 49 25 9 4 7 5 3 2 2 2 2 2 10 ( 2 = + + + = + + + = < x x x prime) is 2 4 1 2 1 1 2 2 2 1 0 0 = + + + = + + = - = - i i

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ics141-lecture10-Summations - University of Hawaii ICS141...

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