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Limit of a sequence

A fundamental question that arises regarding infinite sequences is the behavior of the terms as n gets larger. Since a sequence is a function defined on the positive integers, it makes sense to discuss the limit of the terms as n . For example, consider the following four sequences and their different behaviors as n (see [link] ):

  1. { 1 + 3 n } = { 4 , 7 , 10 , 13 ,… } . The terms 1 + 3 n become arbitrarily large as n . In this case, we say that 1 + 3 n as n .
  2. { 1 ( 1 2 ) n } = { 1 2 , 3 4 , 7 8 , 15 16 ,… } . The terms 1 ( 1 2 ) n 1 as n .
  3. { ( −1 ) n } = { 1 , 1 , −1 , 1 ,… } . The terms alternate but do not approach one single value as n .
  4. { ( −1 ) n n } = { −1 , 1 2 , 1 3 , 1 4 ,… } . The terms alternate for this sequence as well, but ( −1 ) n n 0 as n .
Four graphs in quadrants 1 and 4, labeled a through d. The horizontal axis is for the value of n and the vertical axis is for the value of the term a _n. Graph a has points (1, 4), (2, 7), (3, 10), (4, 13), and (5, 16). Graph b has points (1, 1/2), (2, 3/4), (3, 7/8), and (4, 15/16). Graph c has points (1, -1), (2, 1), (3, -1), (4, 1), and (5, -1). Graph d has points (1, -1), (2, 1/2), (3, -1/3), (4, 1/4), and (5, -1/5).
(a) The terms in the sequence become arbitrarily large as n . (b) The terms in the sequence approach 1 as n . (c) The terms in the sequence alternate between 1 and −1 as n . (d) The terms in the sequence alternate between positive and negative values but approach 0 as n .

From these examples, we see several possibilities for the behavior of the terms of a sequence as n . In two of the sequences, the terms approach a finite number as n . In the other two sequences, the terms do not. If the terms of a sequence approach a finite number L as n , we say that the sequence is a convergent sequence and the real number L is the limit of the sequence. We can give an informal definition here.

Definition

Given a sequence { a n } , if the terms a n become arbitrarily close to a finite number L as n becomes sufficiently large, we say { a n } is a convergent sequence    and L is the limit of the sequence . In this case, we write

lim n a n = L .

If a sequence { a n } is not convergent, we say it is a divergent sequence    .

From [link] , we see that the terms in the sequence { 1 ( 1 2 ) n } are becoming arbitrarily close to 1 as n becomes very large. We conclude that { 1 ( 1 2 ) n } is a convergent sequence and its limit is 1 . In contrast, from [link] , we see that the terms in the sequence 1 + 3 n are not approaching a finite number as n becomes larger. We say that { 1 + 3 n } is a divergent sequence.

In the informal definition for the limit of a sequence, we used the terms “arbitrarily close” and “sufficiently large.” Although these phrases help illustrate the meaning of a converging sequence, they are somewhat vague. To be more precise, we now present the more formal definition of limit for a sequence and show these ideas graphically in [link] .

Definition

A sequence { a n } converges to a real number L if for all ε > 0 , there exists an integer N such that | a n L | < ε if n N . The number L is the limit of the sequence and we write

lim n a n = L o r a n L .

In this case, we say the sequence { a n } is a convergent sequence. If a sequence does not converge, it is a divergent sequence, and we say the limit does not exist.

We remark that the convergence or divergence of a sequence { a n } depends only on what happens to the terms a n as n . Therefore, if a finite number of terms b 1 , b 2 ,… , b N are placed before a 1 to create a new sequence

b 1 , b 2 ,… , b N , a 1 , a 2 ,… ,

this new sequence will converge if { a n } converges and diverge if { a n } diverges. Further, if the sequence { a n } converges to L , this new sequence will also converge to L .

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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