5.1 Sequences  (Page 3/25)

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 $n\to \infty .$ For example, consider the following four sequences and their different behaviors as $n\to \infty$ (see [link] ):

1. $\left\{1+3n\right\}=\{4,7,10,13\text{,…}\}.$ The terms $1+3n$ become arbitrarily large as $n\to \infty .$ In this case, we say that $1+3n\to \infty$ as $n\to \infty .$
2. $\left\{1-{\left(\frac{1}{2}\right)}^n\right\}=\left\{\frac{1}{2},\frac{3}{4},\frac{7}{8},\frac{15}{16}\text{,…}\right\}.$ The terms $1-{\left(\frac{1}{2}\right)}^n\to 1$ as $n\to \infty .$
3. $\left\{{\left(\mathrm{-1}\right)}^n\right\}=\{\text{−}1,1,\mathrm{-1},1\text{,…}\}.$ The terms alternate but do not approach one single value as $n\to \infty .$
4. $\left\{\frac{{\left(\mathrm{-1}\right)}^n}{n}\right\}=\left\{\mathrm{-1},\frac{1}{2},-\frac{1}{3},\frac{1}{4}\text{,…}\right\}.$ The terms alternate for this sequence as well, but $\frac{{\left(\mathrm{-1}\right)}^n}{n}\to 0$ as $n\to \infty .$

From these examples, we see several possibilities for the behavior of the terms of a sequence as $n\to \infty .$ In two of the sequences, the terms approach a finite number as $n\to \infty .$ In the other two sequences, the terms do not. If the terms of a sequence approach a finite number $L$ as $n\to \infty ,$ 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.

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

We remark that the convergence or divergence of a sequence $\left\{a_n\right\}$ depends only on what happens to the terms $a_n$ as $n\to \infty .$ Therefore, if a finite number of terms $b_1,b_2\text{,…},b_N$ are placed before $a_1$ to create a new sequence

this new sequence will converge if $\left\{a_n\right\}$ converges and diverge if $\left\{a_n\right\}$ diverges. Further, if the sequence $\left\{a_n\right\}$ converges to $L,$ this new sequence will also converge to $L.$