Sequence of Real Numbers

Definition

A sequence of real numbers (or sequence in $\mathbb{R}$) is a function with domain $\mathbb{N}$ and codomain $\mathbb{R}$.

Remark

In other words, a sequence in $\mathbb{R}$ assigns to each $n\in \mathbb{N}$ a uniquely determined real number.

It is usually denoted as $X,(x_n)(x_n:n\in \mathbb{N})$

The values of $x_n$ are called $n$th-term or elements of the sequence.

Definining a sequence

A sequence can also be defined by:

1. Formula of $n$-th term:

$$X:=\left(\frac{1}{2n}:n\in \mathbb{N}\right)$$

2. List the terms sequentially, until the pattern seems evident:

$$X:=\left(\frac{1}{2},\frac{1}{4},\frac{1}{6},\frac{1}{8},\dots \right)$$

3. Inductively/recursively

$$x_1:=2,x_{n+1}:=x_n+2$$

4. Constant sequence. e.g., $(0,0,0,\dots )$.