Sequences

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Sequences Summations

• CS 1050
• Rosen 3.2

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Sequence

• A sequence is a discrete structure used to
  represent an ordered list.
• A sequence is a function from a subset of the set
  of integers (usually either the set 0,1,2,. . .
  or 1,2, 3,. . .to a set S.
• We use the notation an to denote the image of the
  integer n. We call an a term of the sequence.
• Notation to represent sequence is an

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Examples

• 1, 1/2, 1/3, 1/4, . . . or the sequence an
  where an 1/n, n?Z .
• 1,2,4,8,16, . . . an where an 2n, n?N.
• 12,22,32,42,. . . an where an n2, n?Z

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Common Sequences
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Summations

• Notation for describing the sum of the terms
  am, am1, . . ., an from the sequence, an
• 
  n
• amam1 . . . an ? aj
• 
  jm
• j is the index of summation (dummy variable)
• The index of summation runs through all integers
  from its lower limit, m, to its upper limit, n.

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Examples
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Summations follow all the rules of multiplication
and addition!
c(12n) c 2c nc
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Telescoping Sums
Example
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Closed Form Solutions
A simple formula that can be used to calculate a
sum without doing all the additions. Example
Proof First we note that k2 - (k-1)2 k2 -
(k2-2k1) 2k-1. Since k2-(k-1)2 2k-1, then
we can sum each side from k1 to kn
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Proof (cont.)
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Closed Form Solutions to Sums
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Double Summations
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Cardinality

• Earlier we defined cardinality of a set as the
  number of elements in the set. We can extend this
  idea to infinite sets.
• The sets A and B have the same cardinality if and
  only if there is a one-to-one correspondence from
  A to B.
• A set that is either finite or has the same
  cardinality as the set of natural numbers is
  called countable. A set that is not countable is
  called uncountable.

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Cardinality

• Cardinality of set of natural numbers?
• An infinite set is countable if and only if it is
  possible to list the elements in a sequence
  (indexed by the positive integers).
• Why? A one-to-one correspondence f can be
  expressed in terms of the sequence a1, a2, a3.,
  where a1 f(1), a2 f(2), etc.
• One-to-one correspondence for set of odd positive
  integers (in terms of positive integers)?
• f(n) 2n - 1