(CSC 102) Lecture 23 Discrete Structures. Previous Lecture Summery Sequences Alternating Sequence ...
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Transcript of (CSC 102) Lecture 23 Discrete Structures. Previous Lecture Summery Sequences Alternating Sequence ...
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(CSC 102)
Lecture 23
Discrete Structures
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Previous Lecture Summery
Sequences
Alternating Sequence
Summation Notation
Product Notation
Properties of Sequences
Change of Variable
Factorial Notations
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Change Of Variable
Observe thet
And also
This equation illustrates the fact that the symbol used to represent the index of a summation can be replaced by any other symbol as long as the replacement is made in each location where the symbol occurs. As a consequence, the index of a summation is called a dummy variable. A dummy variable is a symbol that derives its entire meaning from its local context. Outside of that context (both before and after), the symbol may have another meaning entirely.
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Transforming a Sum by a Change of Variable
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Upper Limit Appears in the Expression to Be Summed
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Cont…
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Factorial Notations
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Examples
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Cont….
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Special Factorial Cases
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Example
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Mathematical Induction I
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Mathematical Induction
Principle of Mathematical Induction
Method of proof
Finding Terms of Sequences
Sum of Geometric Series
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Introduction
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Cont….
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Cont…
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Principal Of Mathematical Induction
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Method Of Proof
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Finding Terms of Sequences
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Explicit formula
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Propositions
Proposition: For all integers n ≥ 1,1 + 2+· · ·+n = n(n + 1)/2
Proof:
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Cont…
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Cont…
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Sum of Geometric Series
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Cont….
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Cont…
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Deducing Additional Formula
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As with the formula for the sum of the first n integers, there is a way to think of the formula for the sum of the terms of a geometric sequence that makes it seem simple and intuitive. Let
Equating the right-hand sides of equations and dividing by r − 1 gives
Cont…
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Lecture Summery
Change of Variable
Factorial Notations
Mathematical Induction
Method of proof
Finding Terms of Sequences
Sum of Geometric Series