Sequences

MATH1051

3 Sequences

A sequence $\{a_n\}$ is an ordered list of numbers

$$a_0,a_1,a_2,a_3..a_n,\dots$$

A sequence can contain a finite number of terms or may continue forever.

3.1 Formal Definition: Sequence

More formally, a sequence is a function, with domain being $\{0,1,2,3,\dots \}$. We can also take the domain as $\{1,2,3,\dots \}$ and start the sequence at $a_1$ rather than $a_0$.

If $a:\{0,1,2,\dots \}\to \mathbb{R}$ is a function, viewed as a sequence, then we write $a_0$ instead of $a(0)$, $a_1$ instead of $a(1)$, etc.

3.2 Representations

There are two main ways to represent the $n^{th}$ term in a sequence. Firstly, there is a direct (or closed form or functional) representation. Secondly, there is a recursive (or indirect) representation.

A direct representation is a formula for $a_n$ in terms of $n$. A recursive description gives a way of obtaining $a_n$ from the previously calculated $a_0,a_1,\dots ,a_{n-1}$.

Often in the recursive case the value of $a_n$ will only depend on the previous 1 or two terms, such as $a_n=f(a_{n-1},a_{n-2})$ or $a_n=g(a_{n-1})$.

3.3 Limits

Let $\{a_n{\}}_{n=0}^{\infty }$ be a sequence. Then

$$\underset{n\to \infty }{\lim }{a}_n=\ell ,\ell \in \mathbb{R}$$

As $a_n$ approaches $\ell$, $n$ gets larger and larger. $a_n$ is always close to $\ell$ for $n$ sufficiently large.

3.3.1 Convention

If a sequence $\{a_n{\}}_{n=0}^{\infty }$ has limit $\ell \in \mathbb{R}$, we say that $a_n$ converges to $\ell$ and that the sequence $\{a_n{\}}_{n=0}^{\infty }$ is convergent. Otherwise the sequence is divergent.

3.4 Theorem: Limit Laws

Caution

The following limit laws apply provided that the separate limits exist (that is $\{a_n\}$ and $\{b_n\}$ are convergent):

Suppose that $\{a_n\}$ and $\{b_n\}$ are convergent sequences such that

$$\underset{n\to \infty }{\lim }{a}_n=\ell \text{ and }\underset{n\to \infty }{\lim }{b}_n=m$$

and $c$ is a constant. Then

$$\begin{array}{rl}\underset{n\to \infty }{\lim }(a_n\pm b_n)& =\ell \pm m;\\ \underset{n\to \infty }{\lim }c{a}_n& =c\ell ;\\ \underset{n\to \infty }{\lim }(a_n{b}_n)& =\ell m;\\ \underset{n\to \infty }{\lim }\frac{a_n}{b_n}& =\frac{\ell }{m}\end{array}$$

3.5 Useful Sequences to Remember

Take care with inequalities and limits. For example $\frac{1}{n}>0$ for all $n$ but ${\lim }_{n\to \infty }\frac{1}{n}=0$. In general, even if $a_n>b_n$ for all $n$, we can only conclude ${\lim }_{n\to \infty }{a}_n\ge {\lim }_{n\to \infty }{b}_n$. Note the $\ge$.

3.6 Theorem: Squeeze

If $a_n\le b_n\le c_n$ for $n\ge n_0$ for some $n_0\in \mathbb{N}$ and ${\lim }_{n\to \infty }{a}_n={\lim }_{n\to \infty }{c}_n=\ell$, then

$$\underset{n\to \infty }{\lim }{b}_n=\ell$$

3.6.1 Example

Use the squeeze theorem on $\{a_n\}$, where $a_n=\frac{1}{n}\sin n$.

Since

$$-1\le \sin n\le 1,\text{ for }n\ge 1,$$

we have

$$-\frac{1}{n}\le a_n\le \frac{1}{n}$$

Now

$$\underset{n\to \infty }{\lim }-\frac{1}{n}=\underset{n\to \infty }{\lim }\frac{1}{n}=0$$

3.7 The Formal Definition of a Limit of a Sequence

We write

$$\underset{n\to \infty }{\lim }{a}_n=\ell$$

if for every number $ϵ>0$ there exists a number $n_0$ such that

$$\mid a_n-\ell \mid <ϵ\text{ whenever }n>n_0$$

For example, ${\lim }_{n\to \infty }{a}_n=0$ means $\mid a_n\mid <ϵ$ whenever $n>n_0$.