Section 2.1 Limits of Sequences

Section 2.1 Limits of Sequences - Section 2.1: Sequences of...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 = ....
View Full Document

Page1 / 5

Section 2.1 Limits of Sequences - Section 2.1: Sequences of...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online