Section 8-1 - Sequences

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Chapter 8: Infinite Sequences and Series

Section 8.1: Sequences

Essentials

Table 8.1.1 summarizes some of the terms and issues that arise in the study of infinite sequences.

Item

Explication

Sequence

•	An infinite sequence is an ordered list of real numbers.

•	A formal definition: A function from the integers to the real numbers

Notation for a Sequence

•	${\{a_{n}\}}_{n = n_{0}}^{\infty}$ , where $a_{n}$ is the general term for the sequence, and $n_{0}$ is the starting index.

•	The notation is often shortened to $\{a_{n}\}$ to save printing costs.

Increasing Sequence

•	A sequence for which $a_{n + 1} \geq a_{n}$ for all $n \geq n_{0}$

Decreasing Sequence

•	A sequence for which $a_{n + 1} \leq a_{n}$ for all $n \geq n_{0}$

Monotone Sequence

•	A sequence that is either increasing or decreasing

Sequence Bounded Above

•	A sequence for which $a_{n} \leq M$ for all $n \geq n_{0}$ and some (finite) real number $M$

Sequence Bounded Below

•	A sequence for which $m \leq a_{n}$ for all $n \geq n_{0}$ and some (finite) real number $m$

Bounded Sequence

•	A sequence that is both bounded above and below

Limit of a Sequence

•	Informally: The numbers $a_{n}$ approach the number $L$ as $n \to \infty$ .

•	Formally: The sequence $\{a_{n}\}$ has limit $L$ , that is, $lim_{n \to \infty} a_{n} = L$ , if for each $ϵ > 0$ there is an integer $N$ for which $\|a_{n} - L\| < ϵ$ for all $n > N$ .

Convergent Sequence

•	If $lim_{n \to \infty} a_{n} = L$ , where $L$ is a finite real number, then the sequence $\{a_{n}\}$ is said to converge.

Divergent Sequence

•	If $lim_{n \to \infty} a_{n} = \infty$ or does not exist (because of oscillation), then the sequence $\{a_{n}\}$ is said to diverge.

Table 8.1.1 Key terms related to infinite sequences

Table 8.1.2 lists two useful limits for sequences. If the index were a continuous variable, these limits could be obtained with the tools developed in Chapter 1.

$lim_{n \to \infty} \frac{1}{n^{r}} = 0$ for $r > 0$

$lim_{n \to \infty} r^{n} = {\begin{array}{c} 0 & |r| < 1 \\ 1 & r = 1 \end{array}$

Table 8.1.2 Two useful limits

Theorem 8.1.1 states that if $f \to L$ as the continuous variable $x$ becomes infinite, then certainly the values of $f$ at the integers must also approach $L$ . Theorem 8.1.1 applies to those sequences whose general term $a_{n}$ is the value of some function $f$ at the integer $n$ . In other words, Theorem 8.1.1 makes a statement about the limit of the sequence ${\{f (n)\}}_{n = n_{0}}^{\infty}$ . If $f (x) \to L$ , then $f (n) \to L$ also.

Theorem 8.1.1

If $lim_{x \to \infty} f (x) = L$ , and $a_{n} = f (n)$ , then $lim_{n \to \infty} a_{n} = lim_{n \to \infty} f (n) = L$ .

Hence, Theorem 8.1.1 permits the application of the limit properties in Table 1.3.1, and any of the theory of indeterminate forms in Section 3.9 to be applied to sequences whose general term can be interpreted as the value of a function. In fact, Theorem 8.1.1 even permits the application of L'Hôpital's rule for such sequences.

Theorem 8.1.2

A bounded monotone sequence converges.

Intuitively, it is appealing that a sequence that is either increasing or decreasing, and that is also bounded, should converge. However, Theorem 8.1.2 is really a deep statement about the real numbers: it declares that there is actually a real number $L$ to serve as the limit of the sequence. Hence, this is as much a statement about the completeness of the real numbers as it is about the behavior of certain sequences.

Theorem 8.1.3: The Discrete Squeeze Theorem

1.	$a_{n} \leq b_{n} \leq c_{n}$ for all $n > N$ for some $N$

2.	$a_{n} \to L$ and $c_{n} \to L$ as $n \to \infty$

⇒

1.	$b_{n} \to L$

Theorem 8.1.3 is the discrete analog of the Squeeze theorem stated for continuous functions in Section 1.3.

Examples

Example 8.1.1

If $a_{n} = \frac{2 n^{2} - 3 n}{4 n^{3} + 5}$ , find the limit of the sequence ${\{a_{n}\}}_{n = 0}^{\infty}$ .

Example 8.1.2

If $a_{n} = \frac{\ln (n)}{n}$ , find the limit of the sequence ${\{a_{n}\}}_{n = 1}^{\infty}$ .

Example 8.1.3

If $b_{n} = \frac{{(- 2)}^{- n}}{n}$ , find the limit of the sequence ${\{b_{n}\}}_{n = 1}^{\infty}$ .

Example 8.1.4

If $a_{n} = \frac{n &excl;}{n^{n}}$ , find the limit of the sequence ${\{a_{n}\}}_{n = 1}^{\infty}$ .

Example 8.1.5

If $a_{n} = \frac{n}{n^{2} + 1}$ , show the sequence ${\{a_{n}\}}_{n = 0}^{\infty}$ is decreasing.

Example 8.1.6

If $a_{n} = \frac{5^{n}}{n &excl;}$ , find the limit of the sequence ${\{a_{n}\}}_{n = 0}^{\infty}$ .

Example 8.1.7

If $a_{n} = \frac{1 \cdot 3 \cdot 5 \cdot \dots \cdot (2 n - 1)}{n^{n}}$ , show that the sequence ${\{a_{n}\}}_{n = 1}^{\infty}$ is decreasing.

Example 8.1.8

If $a_{1} = 1, a_{2} = - 1$ , and $6 a_{n + 2} - 5 a_{n + 1} + a_{n} = 0$ defines $a_{n}$ for $n > 2$ , use Maple to find the general term $a_{n}$ .

Example 8.1.9

a)	If $a_{1} = 1$ and $a_{n + 1} = \frac{4}{3 + 2 a_{n}}$ for $n > 1$ , graph $a_{n}$ for $n = 1, \dots, 10$ .

b)	Use Maple to find an explicit representation for $a_{n}$ .

c)	Use Maple to calculate $lim_{n \to \infty} a_{n}$ .

Example 8.1.10

If $a_{n} = \frac{5 n - 4}{2 n + 3}$ , determine if the sequence ${\{a_{n}\}}_{n = 0}^{\infty}$ converges or diverges.