Sequence: definition and notation

Before studying the limit of a function of a real variable, it is convenient to work with a simpler and more visual case: the limit of a sequence. A sequence is an infinite ordered list of real numbers; the typical questions are “what number do the terms of the list approach? do they approach a precise value or do they grow indefinitely?”. The answer to these questions is the first, fundamental, step towards analysis.

Definition — Sequence

A sequence of real numbers is a function $a:\mathbb{N}\to\mathbb{R}$. The value at $n$ is denoted by $a_n$ (instead of $a(n)$) and is called the $n$-th term of the sequence. The sequence is denoted $(a_n)_{n\in\mathbb{N}}$ or simply $(a_n)$.

Example

• $a_n = n$:   $0, 1, 2, 3, \ldots$ (sequence of the natural numbers).
• $a_n = 1/n$ for $n\ge 1$:   $1, 1/2, 1/3, 1/4, \ldots$
• $a_n = (-1)^n$:   $1, -1, 1, -1, \ldots$ (oscillating).
• $a_n = 2^n$:   $1, 2, 4, 8, 16, \ldots$ (geometric with common ratio $2$).

Links

Topics: Sequences
Concepts: Sequence · General term