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Chapter 22 - Sequence and summation. In this chapter, the following content will be discussed: Sequences, alternating sequence, summation notation, product notation, properties of summation, change of variable, factorial notations. | Sequence and Summation (CSC 102) Lecture 22 Discrete Structures Today’s Lecture Sequences Alternating Sequence Summation Notation Product Notation Properties of Summation Change of Variable Factorial Notations Sequences A sequence is a function whose domain is either all the integers between two given integers or all the integers greater than or equal to a given integer. We typically represent a sequence as a set of elements written in a row. In the sequence denoted am, am+1, am+2, . . . , an each individual element ak (read “a sub k”) is called a term. The k in ak is called a subscript or index, m. Definition Finding Terms of Sequences Compute the first six terms of the sequence c0, c1, c2, . . . defined as follows: c j = (−1) j for all integers j ≥ 0. Thus the first six terms are 1,−1, 1,−1, 1,−1. It follows that the sequence oscillates endlessly between 1 and −1. Alternating Sequence Solution: Denote the general term of the sequence by ak and suppose the first term is a1.Then observe that the denominator of each term is a perfect square. Thus the terms can be rewritten as: Note that the denominator of each term equals the square of the subscript of that term, and that the numerator equals ±1. Hence Explicit formula Cont . Also the numerator oscillates back and forth between +1 and −1; it is +1 when k is odd and −1 when k is even. To achieve this oscillation, insert a factor of (−1)k+1 (or (−1)k−1) into the formula for ak . [For when k is odd, k + 1 is even and thus (−1)k+1 = +1; and when k is even, k + 1 is odd and thus (−1)k+1 = −1.] Consequently, an explicit formula that gives the correct first six terms is Note that making the first term a0 would have led to the alternative formula Summation Cont Summation Notation to expanded form Expanded form to Summation Notation A more mathematically precise definition of summation, called a recursive definition, is the following: If m is any integer, then Hence When solving problems, it is often useful to rewrite a . | Sequence and Summation (CSC 102) Lecture 22 Discrete Structures Today’s Lecture Sequences Alternating Sequence Summation Notation Product Notation Properties of Summation Change of Variable Factorial Notations Sequences A sequence is a function whose domain is either all the integers between two given integers or all the integers greater than or equal to a given integer. We typically represent a sequence as a set of elements written in a row. In the sequence denoted am, am+1, am+2, . . . , an each individual element ak (read “a sub k”) is called a term. The k in ak is called a subscript or index, m. Definition Finding Terms of Sequences Compute the first six terms of the sequence c0, c1, c2, . . . defined as follows: c j = (−1) j for all integers j ≥ 0. Thus the first six terms are 1,−1, 1,−1, 1,−1. It follows that the sequence oscillates endlessly between 1 and −1. Alternating Sequence Solution: Denote the general term of the sequence by ak and suppose the first term is a1.Then .
