lecture13 - Lecture 13 Summations n ai im am am 1 . an Some...

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Lecture 13
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Summations n m i n m m i a a a a ... 1
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Some important summation formulae 2 ) 1 ( 1 n n i n i
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Some important summation formulae 1 , 0 , 1 1 1 0 r r r r n n i i
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When |x| < 1Taking the limit of the previous one gives 1 | | , 1 1 0 x x x i i
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Other common notation Double summation 1 0 2 1 ) 2 1 ( ) 1 1 ( ) 2 0 ( ) 1 0 ( ) ( i j j i
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Set notation S = {3,5,7} S x x 7 5 3
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Cardinality revisited Cardinality: number of elements Can we extend this notion to infinite sets? Sets A and B have the same cardinality iff there is a 1-1 correspondence between A and B
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Countable sets Finite sets are countable A set that has the same cardinality as the set of positive integers is countable Also called enumerable
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Example Show that the set of positive even numbers is countable Use the mapping f(n) = 2n Need to show that f is 1-1 and onto 1-1: Suppose f(n) = f(m). Then 2n = 2m and n =m Onto: Take any integer even x. x = 2k f(k) = x
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All integers Show that the set of all integers is countable …, -2, -1, 0, 1, 2, … reorder 0,1,-1,2,-2,… odd is n when 2 / ) 1 ( even n when 2 / ) ( n n n f
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lecture13 - Lecture 13 Summations n ai im am am 1 . an Some...

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