Chapter3 - Chapter 3 Sequences and Their Limits 3.0...

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Chapter 3 Sequences and Their Limits 3.0 Definition of Sequence Informally, a sequence is an infinite “list” of numbers where order is important. For example, { 0 , 1 , 0 , 1 , 0 , 1 , . . . } is a sequence. Formally, we make the following definition. Definition 3.0.1. A sequence is a function f : N R . For n N , we call f ( n ) the n th term of the sequence; for the sake of intuitive notation, we will usually denote the n th term of the sequence by x n , y n , a n , b n , or some other letter with n as subscript, instead of f ( n ). We will denote the whole sequence by { x n } n N or simply by { x n } , where x could be any letter. Note that a sequence does not need to have a pattern at all; however, most of the sequence we will be working with will be defined by formulae such as 1 n . When we want to write down a sequence, we can specify how to compute its n th term, as in 1 n , or we can list the terms until the pattern becomes apparent, as in 1 , 1 2 , 1 3 , 1 4 , . . . . We will frequently encounter sequences that are defined by recursion . An example of such a sequence is: a 1 = 1 , a n +1 = 3 + 2 a n . You wil notice that in the above example we can use what we kow of the first term to calculate the second term. We can then determine the third term, and so on. For recursively defined sequences calculating a particular term requires us to already know the values of previous terms. Diversion . Is there a difference between induction and recursion? Given a sequence { a 1 , a 2 , a 3 , . . . } , a tail of the sequence is a sequence of the form { a k , a k +1 , a k +2 , . . . } ; that is, a collection of terms in the original sequence 24
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from some fixed point onwards—thus the word tail . A tail is obtained by deleting the first few terms off the sequence. 3.1 Limits of Sequences Defining the limit. Consider the sequence 1 , 1 2 , 1 3 , . . . , 1 n , . . . or 1 n n N . What can we say about the a n ’s as n gets very large? As n gets larger, these germs are getting “closer and closer” to the value 0. We say that 0 is the limit of the sequence { a n } as n → ∞ , or { a n } converges to 0. We give below an informal, heuristic definition of convergence. Informal definition . Given a sequence { a n } , we say that L is the limit of { a n } as n goes to infinity if as n gets very large, the a n ’s get “closer and closer” to L . There is a problem with this definition. Note that as n gets larger, a n = 1 n gets closer and closer to - 17; however, we do not want - 17 to be a limit! What we are missing is a way of quantifying the fact that as n gets large all of the terms in the tail of the sequence are very close to 0. The following is the formal definition of limit. Definition 3.1.1 [Limit] . We say that L is the limit of a sequence { a n } (notation: lim n →∞ a n = L ) if for every > 0, these exists an N N (which in general depends on ) such that n N implies | a n - L | < .
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