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Unformatted text preview: Chapter 4 Sequences and SeriesFirst View Recall that, for any set A , a sequence of elements of A is a function f : M A . Rather than using the notation f n for the elements that have been selected from A , since the domain is always the natural numbers, we use the notational convention a n f n and denote sequences in any of the following forms: a n * n 1 a n n + M or a 1 a 2 a 3 a 4 This is the only time that we use the set bracket notation in a different con- text. The distinction is made in the way that the indexing is communicated . For a n : , the a n * n 1 is the constant sequence that lists the term : in nitely often, : : : : while a n : n + M is the set consisting of one element : . (When you read the last sentence, you should have come up with some version of For a sub n equal to : , the sequence of a sub n for n going from one to in nity is the constant sequence that lists the term : in nitely often, : : : while the set consisting of a sub n for n in the set of positive integers is the set consisting of one element : i.e., the point is that you should not have skipped over the a n * n 1 and a n : n + M .) Most of your previous experience with sequences has been with sequences of real numbers, like 1 1 2 3 5 8 13 21 34 55 ,... | 3 n 1 } * n 1 | n 2 3 n 5 n 47 } * n 1 | n 3 1 n 3 1 1 n } * n 1 and | log n n sin r n H 8 s } * n 1 . In this chapter, most of our sequences will be of elements in Euclidean n-space. In MAT127B, our second view will focus on sequence of functions. 123 124 CHAPTER 4. SEQUENCES AND SERIESFIRST VIEW As children, our rst exposure to sequences was made in an effort to teach us to look for patterns or to develop an appreciation for patterns that occur naturally. Excursion 4.0.1 For each of the following, nd a description for the general term as a function of n + M that ts the terms that are given. 1. 2 5 4 7 8 9 16 11 32 13 64 15 2. 1 3 5 9 7 9 81 11 13 729 *** An equation that works for (1) is 2 n 2 n 3 1 while (2) needs a different for- mula for the odd terms and the even terms one pair that works is 2 n 1 2 n 1 1 for n even and 3 n 1 when n is odd. ** * As part of the bigger picture, pattern recognition is important in areas of math- ematics or the mathematical sciences that generate and study models of various phenomena. Of course, the models are of value when they allow for analysis and/or making projections. In this chapter, we seek to build a deeper mathematical under- standing of sequences and series primary attention is on properties associated with convergence. After preliminary work with sequences in arbitrary metric spaces, we will restrict our attention to sequences of real and complex numbers....
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