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Lecture Discrete structures: Chapter 23 - Amer Rasheed

In this chapter, the following content will be discussed: Change of variable, transforming a sum by a change of variable, upper limit appears in the expression to be summed, special factorial cases, mathematical induction I. | (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 . | (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