Definition and notation of a sequence

What is a numerical sequence?

A numerical sequence is an ordered, infinite list of numbers, indexed by the natural numbers. Each number in the list is called a term, and its position in the list is called its rank (or index). A sequence can be viewed as a function which, for each integer n, assigns a real number denoted by u_n (and not u(n), even though the idea is similar).

Notation

We denote the sequence as (u_n), and each individual term as u_0, u_1, u_2, …, u_n, u_(n+1), etc. The first term is not always u_0: depending on the context, a sequence may start at u_1 or even at a different index. It is therefore always necessary to check the starting index before calculating anything.

Example

Let $u_n = 2n + 1$ for all integers $n \geq 0$.

n	0	1	2	3
u_n	1	3	5	7

We obtain u_0 = 1, u_1 = 3, u_2 = 5, u_3 = 7: the terms increase regularly by 2.

Graphical representation

A sequence is often represented by a scatter plot of isolated points (n; u_n) in a coordinate system, never by a continuous curve: a sequence is defined only for integers, not for intermediate values such as n = 1.5.

Common pitfall

Do not confuse the index n (an integer, a position) with the term u_n (a number, a value). Another common mistake is forgetting to check from which index the sequence is defined before calculating u_0 or computing a sum.