2.2. Sequences

In analysis, we often deal with functions defined on continuous domains. For example, a cubic function defined on the real number set \(\mathbb{R}\) as

\[ f(x)=x^3, \qquad x \in \mathbb{R}. \]

Sometimes the domain might not be continuous but discrete. For example, we can define an exponential function, with base \(3\), on the integer number set \(\mathbb{Z}\) as

(2.6)\[ f(n)=3^n, \qquad n \in \mathbb{Z}, \]

where \(\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}\).

We can also define a quadratic function on the whole number set \(\mathbb{N}_0\) as

(2.7)\[ f(n)=n^2, \qquad n \in \mathbb{N}_0, \]

where \(\mathbb{N}_0=\{0, 1, 2, 3,\ldots\}\), and so the output of this function will be

\[ 0, 1, 4, 9, 16, \ldots \]

Functions given in (2.6) and (2.7) are called sequences.

Definition 2.5 (Sequence)

In mathematics, a sequence is a function whose domain is a set of integers. An output of a sequence is called a member or term of it. Specifically, we will regard the expression \(\{y_n\}_{n=0}^{+\infty}\) to be an alternative notation for the function \(y_n=y(n), n=0,1,2,\ldots\)

Note

A sequence is not necessarily a set. Elements in a set cannot be repeated, while members in a sequence can be repeated, e.g. \(0, 1, 1, 2, 2, 2, 3.7, 4.9, 4.9,\ldots\)