# Sequences

### Definition

A sequence $$\{a_n\}$$ has the limit $$L$$ if
for every $$\epsilon >0$$ $$~~ \exists ~~N \backepsilon$$
if $$~~n > N$$
then
$$~~ |a_n - L| < \epsilon$$

### Properties of Limits of Sequences

These properties are the same as the properties of limits of functions

Theorem:

Assume that the sequence $$\{a_n\}$$ has a limit $$A$$ and the sequence $$\{b_n\}$$ has a limit $$B$$.

Then,

• $$\lim_{n\to\infty}\limits (a_n\pm b_n) = A \pm B$$
• $$\lim_{n\to\infty}\limits (c a_n) = cA$$
• $$\lim_{n\to\infty}\limits (a_n b_n) = A B$$
• $$\lim_{n\to\infty}\limits (\frac{a_n }{b_n}) = \frac AB$$ provided $$B\neq 0$$

### Example:

The sequence $$\left\lbrace a_n \right\rbrace=\left\lbrace \frac{4n+3}{n} \right\rbrace$$ has the limit $$4$$ and the sequence $$\left\lbrace b_n \right\rbrace=\left\lbrace \frac{2n}{8n-1} \right\rbrace$$ has the limit $$\frac{1}{4}$$

1. Find $$\lim_{n\to\infty}\limits \left(\frac{a_n }{b_n}\right)$$
2. Find $$\lim_{n\to\infty}\limits (a_n b_n)$$
3. Find $$\lim_{n\to\infty}\limits \left(\frac{4n+3}{3n} \right)$$

### Limits of Sequences as Functions

Theorem:
Suppose $$f$$ is a function such that $$f(n) = a_n$$ for all positive integers $$n$$.
If $$\lim_{x\to\infty}\limits f(x)=L$$, then the limit of the sequence $$\{a_n\}$$ is also $$L$$.
Definition:

$$\lim_{n\to\infty} \limits a_n=\infty$$ means that for every positive number $$M$$ there is an integer $$N$$ such that if $$n>N$$ then $$a_n > M$$

### Example:

Find the limit of the sequence $$\left \{\frac{\sin(n)}{n}\right \}$$

## Two useful Theorems:

### Squeeze Theorem

If $$a_n \leq b_n \leq c_n$$ and $$\displaystyle \lim_{n\to \infty}{a_n} = \lim_{n\to \infty}{c_n} = L$$, then $$\displaystyle \lim_{n\to \infty} b_n=L$$

### Absolute Value Theorem

If $$\lim_{n\to \infty} |a_n| = 0$$, then $$\lim_{n\to \infty} a_n = 0$$

### Example:

Evaluate $$\displaystyle \lim_{n\to\infty} \frac{(-1)^n}{n}$$ if it exists

$$\displaystyle \lim_{n\to\infty} \left \vert \frac{(-1)^n}{n} \right \vert = \lim_{n\to\infty} \frac{1}{n} =0$$ So

$$\displaystyle \lim_{n\to\infty} \frac{(-1)^n}{n} =0$$

### Example:

Use the squeeze theorem to evaluate $$\displaystyle \lim_{n\to\infty} \frac{\sin(n)}{n}$$ if it exists

### Note

The geometric sequence $$\{r^n\}$$ is convergent if $$-1 \lt r \leq 1$$ and divergent for all other values of $$r$$.

Futhermore, $\lim_{n\to\infty} = \begin{cases}0 & \text{ if } -1\lt r \lt 1 \\ 1 & \text{ if } r = 1\end{cases}$

### Defnition

A sequence $$\{a_n\}$$ is increasing if $$a_n \lt a_{n+1}$$

A sequence $$\{a_n\}$$ is decreasing if $$a_n \gt a_{n+1}$$

A sequence $$\{a_n\}$$ is monotonic if it is increasing or decreasing

### Definition

A sequence $$\{a_n\}$$ is bounded above if there exists a number $$M$$ such that

$a_n \leq M ~~\forall~ n \geq 1$

A sequence $$\{a_n\}$$ is bounded above if there exists a number $$m$$ such that

$a_n \geq m ~~\forall~ n \geq 1$

A sequence $$\{a_n\}$$ is bounded if it is bounded above and below

### Theorem

Every bounded, monotonic sequence is convergent.

### Example

Determine whether or not the sequence $$\{e^{1/n}\}$$ converges.

### Growth Rates of Functions

There exists an $$N$$ such that if $$n>N$$, the following is true

$\ln^p(n) \lt n^q \lt n^q \ln^r(n) \lt n^{q+s} \lt b^n \lt n! \lt n^n$

By jkesler

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