Sequence¶

Definition¶

Given an integer $m$ (usually 0 or 1), a sequence $(a_n)_{n\geq m}$ is an ordered collection of elements $a_n$ (e.g., numbers, vectors, functions):

a_{m},\,a_{m+1},\,a_{m+2},\,\dots.

Interpretation¶

A sequence is an ordered list indexed by integers: for every index $n\ge m$ there is exactly one term $a_n$ , and the list $a_m,a_{m+1},a_{m+2},\ldots$ never ends. Order distinguishes sequences from sets.

The terms (-1) to the n over n for n = 1 to 10, alternating in sign and approaching 0, one value for each index.

The figure plots $a_n=(-1)^n/n$ for $n=1,\dots,10$ $1, \dots, 10$ , with the index $n$ along the horizontal axis and the value $a_n$ upwards: $a_1=-1$ $- 1$ , $a_2=\tfrac{1}{2}$ $\frac{1}{2}$ , $a_3=-\tfrac{1}{3}$ $- \frac{1}{3}$ , and so on. The terms alternate in sign while their distance to $0$ , $|a_n|=\tfrac{1}{n}$ $\frac{1}{n}$ , shrinks.

Two notations should be kept apart: $a_n$ is a single term, a number, while $(a_n)_{n\ge m}$ is the whole sequence. Passing to the set of values loses information: the sequence $(1,-1,1,-1,\ldots)$ from the Examples has infinitely many terms, but its set of values $\{-1,1\}$ has only two elements. What happens to $a_n$ for large $n$ is the subject of the limit of a sequence.

Key Results¶

A sequence $(a_n)_{n\ge m}$ is a function $n\mapsto a_n$ defined on the subset $\{m,m+1,m+2,\ldots\}$ of the integers.
Order matters in sequences.
A sequence never ends; an ordered list with only finitely many entries is a finite sequence.

Examples¶

$(a_n)_{n=1}^\infty$ denotes an infinite sequence.
The sequence $(1,\,-1,\,1,\,-1,\,\dots)$ oscillates.

Prerequisites¶

Integer

Sources¶

Math I: Calculus · Lecture 2

Symbols: infinity

Last updated: 20 October 2026 03:02