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Unformatted text preview: Section 2.1: Sequences of Real Numbers Now that we are familiar with some of the properties of real numbers, we are ready to turn our attention to the notion of a limit. When speaking about a limit, we may be referring to a sequential limit, the limit of sequence, or a functional limit, the limit of a function at a given point. In this section, we deal with the former type. Understanding the convergence of sequences is essential to a study of analysis, and we will often return to sequential limits in our discussion of functional limits and continuity. Definition: A sequence is a function f whose domain is N , the set of positive integers. We usually denote a sequence by ( ) n a , where ( ) n f n a = is the nth term of the sequence. Unless stated otherwise, we will be working with sequences of real numbers , that is, for every positive integer n , n a R . Many of the theorems we will prove also hold for C , the set of complex numbers, but it is easier to remain in a more familiar framework. Example: Consider the sequence ( ) n a defined by 1 n n a = for each positive integer n . Then the first five terms of the sequence are 1 1 a = , 1 2 2 a = , 1 3 3 a = , 1 4 4 a = , 1 5 5 a = , Intuitively, we know that as n increases, ( ) n a tends to zero. We would like to say that 0 is the limit of the sequence. Before we can, we must make this notion of tends to or converges to more precise. Definition: A real sequence ( ) n a converges to a real number L if for every , there exists a positive integer N such that for every n N , we have n a L - < . If ( ) n a converges to L , we say that L is the limit of the sequence and write lim n a L = , or more commonly, ( ) n a L . If ( ) n a does not converge, then it diverges . Essentially, this definition states that if ( ) n a L , then for any given positive number , we can make it so that for all n sufficiently large, the distance between n a and L is smaller than . Return to the sequence in the previous example defined by 1 n n a = . Suppose we were given a challenge of 1 100 0.01 = = . If ( ) n a truly converges to zero, we should be able to find N so that for all n N , 1 1 100 n- < . To see that this is indeed the case, note that 1 1 n n- = . Therefore, our task simplifies to finding all n such that 1 1 100 n < . We conclude that as long as 100 n , the desired result would follow; with this in mind, take 101 N = ....
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