Calculus/Definition of a Sequence

Finite Sequences

Definition: Definition of a Sequence

A sequence is an ordered collection of terms in which repetition is allowed. The number of terms in a sequence is called the length of the sequence.

Sequences are often denoted by brackets like $\left\{2,3,3,4,4\right\}$. Furthermore if we have a sequence $a$ such that $a=\left\{1,2,3,4,5\right\}$ then $a_{1}=1,a_{2}=2,a_{3}=3...$. The subscript must be a non-negative integer. Also notice that $n$ starts from one and counts up.

We can describe the terms in this sequence with a formula $a_{n}=n$ for all non-negative integers $n<6$. So under this definition $a_{6}$ is not defined, and indeed $a_{6}$ is not in the sequence.

Infinite sequences

Definition: Definition of an infinite sequence

An infinite sequence is a sequence with an infinite number of elements.

Infinite sequences have infinite terms. For such a sequence, we can again give a formula for any term in the sequence. For our previous sequence $a$, we can say $a_{n}=n$ for all non-negative integers $n$. This sequence could also be denoted as $\left\{0,1,2,3,4,5,...\right\}$ where the period of ellipses implies that this sequence is infinite.

Discrete Functions

Earlier, we defined the members of the infinite sequence $a$ as $a_{n}=n$ for all non-negative integers $n$. This is known as a discrete function, discrete definition, or explicit definition. A discrete function is any function whose domain is not the set of all real or imaginary numbers, but is instead a smaller, countable set like the set of all integers or the set of all rational numbers. Note that a set differs from a sequence, but that is beyond the scope of this discussion.

Discrete functions only take “countable”, discrete domains. The set of all integers is countable, because there are not infinitely many values between two values in the set; there is no extra value between 2 and 1, as 1.5 is not an integer and is not contained in the set. Also note that given a discrete function or explicit definition, as long as the domain is discrete, the range must also be discrete. This means that if the input of a discrete function is countable, the output must also be countable.

Example 1

$q_{n}=n+1$

$q=\left\{2,3,4,5,6...\right\}$

This is known as an arithmetic sequence. These will be discussed later.

Example 2

$c_{n}=\cos(n-1)$

$c=\left\{1,0.5403...,-0.4161...,-0.9899...,0.2836...,...\right\}$

This result may be interesting: a sequence does not need to be a collection of integers, indeed it can be any collection, as long as it is countable. Here, we are simply taking the cosine of all integers, and any discrete function must have both a discrete domain and range.

Recursive Functions

Recursive functions, recursive formulas, or recursive definitions are formulas in which $a_{n}$ is defined in terms of $a_{n-1}$. Knowing any term in a recursively defined sequence requires you to know all the terms before it, which means you must know the first term, sometimes denoted $a_{0}$ or $a_{1}$. The first term must be defined in order to have a proper recursive sequence; it cannot be assumed that the first term is 1.

Sometimes, one can have a sequence that is necessarily defined by a recursive function. For instance, the recursively defined sequence $u_{n+1}=\cos(u_{n}),a_{1}=1$. This sequence cannot be expressed any other "easy" way and in this kind of situation it is best to use the recursive definition.

Example 1

The sequence

$p_{n+1}=p_{n}+1,p_{1}=2$

$p=\left\{2,3,4,5...\right\}$

is the same arithmetic sequence mentioned earlier. However, this time it uses a recursive definition which is essentially the same.

Example 2

This is the sequence of cosine mentioned earlier:

$u_{n+1}=\cos(u_{n}),a_{1}=1$

$u=\left\{0.5403...,-0.9111...,0.6128...,0.8180...,...\right\}$

Example 3

$s_{n}=3\times s_{n-1}$

Notice that this time, instead of saying $s_{n+1}=...$, we defined $s_{n}$ in terms of $s_{n-1}$. This definition is still valid.