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(CSC 102)
Lecture 23
Discrete Structures
Previous Lecture Summery
Sequences
Alternating Sequence
Summation Notation
Product Notation
Properties of Sequences
Change of Variable
Factorial Notations
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.
Transforming a Sum by a Change of Variable
Upper Limit Appears in the Expression to Be Summed
Cont…
Factorial Notations
Examples
Cont….
Special Factorial Cases
Example
Mathematical Induction I
Mathematical Induction
Principle of Mathematical Induction
Method of proof
Finding Terms of Sequences
Sum of Geometric Series
Introduction
Cont….
Cont…
Principal Of Mathematical Induction
Method Of Proof
Finding Terms of Sequences
Explicit formula
Propositions
Proposition: For all integers n ≥ 1,1 + 2+· · ·+n = n(n + 1)/2
Proof:
Cont…
Cont…
Sum of Geometric Series
Cont….
Cont…
Deducing Additional Formula
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…
Lecture Summery
Change of Variable
Factorial Notations
Mathematical Induction
Method of proof
Finding Terms of Sequences
Sum of Geometric Series