22c:19 discrete math sequence and sums
DESCRIPTION
22C:19 Discrete Math Sequence and Sums. Fall 2011 Sukumar Ghosh. Sequence. A sequence is an ordered list of elements. . Examples of Sequence. Examples of Sequence. Examples of Sequence. Not all sequences are arithmetic or geometric sequences. An example is Fibonacci sequence. - PowerPoint PPT PresentationTRANSCRIPT
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22C:19 Discrete MathSequence and Sums
Fall 2011Sukumar Ghosh
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SequenceA sequence is an ordered list of elements.
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Examples of Sequence
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Examples of Sequence
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Examples of Sequence
Not all sequences are arithmetic or geometric sequences.An example is Fibonacci sequence
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Examples of Sequence
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More on Fibonacci Sequence
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Examples of Golden Ratio
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Sequence Formula
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Sequence Formula
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Some useful sequences
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Summation
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Evaluating sequences
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Arithmetic Series
Consider an arithmetic series a1, a2, a3, …, an. If the common
difference (ai+1 - a1) = d, then we can compute the kth term ak
as follows:
a2 = a1 + d
a3 = a2 + d = a1 +2 d
a4 = a3 + d = a1 + 3d
ak = a1 + (k-1).d
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Evaluating sequences
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Sum of arithmetic series
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Solve this
Calculate 12 + 22 + 32+ 42 + … + n2
[Answer n.(n+1).(2n+1) / 6]
1 + 2 + 3 + … + n = ?
[Answer: n.(n+1) / 2] why?
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Can you evaluate this?
Here is the trick. Note that
Does it help?
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Double Summation
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Sum of geometric series
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Sum of infinite geometric series
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Solve the following
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Sum of harmonic series
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Sum of harmonic series
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Book stacking example
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Book stacking example
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Useful summation formulae
See page 157 of Rosen Volume 6
orSee page 166 of Rosen Volume
7
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Products
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Dealing with Products
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Factorial
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Factorial
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Stirling’s formula
A few steps are omitted here
Here ~ means that the ratio of the two sides approaches 1 as n approaches ∞
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Countable sets
Cardinality measures the number of elements in a set.
DEF. Two sets A and B have the same cardinality, if and only if there is a one-to-one correspondence from A to B.
Can we extend this to infinite sets?
DEF. A set that is either finite or has the same cardinality as the set of positive integers is called a countable set.
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Countable sets
Example. Show that the set of odd positive integers is countable.
f(n) = 2n-1 (n=1 means f(n) = 1, n=2 means f(n) = 3 and so on)
Thus f : Z+ {the set of of odd positive integers}.
So it is a countable set.
The cardinality of an infinite countable set is denoted by
(called aleph null)
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Countable sets
Theorem. The set of rational numbers is countable.
Why? (See page 173 of the textbook)
Theorem. The set of real numbers is not countable.
Why? (See page 173-174 of the textbook).