# Sequences
Source: algebrica.org - CC BY-NC 4.0
https://algebrica.org/sequences/
Fetched from algebrica.org post 13906; source modified 2026-04-13T21:08:05.
## What is a sequence
A sequence is an ordered collection of elements, each assigned to a specific position indexed by a [natural number](../natural-numbers.md). Let us consider the set of real numbers \\(\\mathbb{R}\\). A sequence with values in \\(\\mathbb{R}\\) is a function of the form \\(\\mathbb{N} \\rightarrow \\mathbb{R}\\), that assigns to each \\( n \\in \\mathbb{N} \\) a unique real number \\( a(n) \\in \\mathbb{R} \\).
- A sequence \\( a : \\mathbb{N} \\rightarrow \\mathbb{R} \\) is denoted by \\(\\lbrace a\_n \\rbrace\_{n \\in \\mathbb{N}}.\\)
- Each element produced by the sequence is known as a term.
- The expression for \\( a\_n \\) defines the rule that determines every term of the sequence.
It is often useful to consider sequences defined only on a subset of natural numbers, such as those starting from a specific [integer](../integers.md) value. These are sequences of the form:
\\[
a : {n \\in \\mathbb{N} : n \\geq n\_0} \\to \\mathbb{R}.
\\]
This means the sequence is defined for all natural numbers greater than or equal to some initial index \\( n\_0 \\).
* * *
Consider, for example, the [function](../functions.md) \\( a: \\mathbb{N}^+ \\to \\mathbb{R} \\) defined by \\( a(n) := \\dfrac{1}{n} \\). This is a real-valued sequence defined for every \\( n \\in \\mathbb{N}^+ \\), and its terms are:
\\[
a\_1 = 1, \\quad a\_2 = \\frac{1}{2}, \\quad \\dots, \\quad a\_n = \\frac{1}{n} \\quad \\forall n \\in \\mathbb{N}^+.
\\]
* * *
Another example of a sequence is \\( a\_n = n! \\), the [factorial](../factorial.md) of \\( n \\), which is defined as the product of all positive integers from 1 to \\( n \\). The first few terms of the sequence are:
\\[
a\_1 = 1, \\quad a\_2 = 2, \\quad a\_3 = 6, \\quad a\_4 = 24, \\quad a\_5 = 120, \\quad \\dots
\\]
## Example
Consider, for example, the formula:
\\[
a\_n := \\frac{1}{n - 2}
\\]
defines a real-valued sequence \\( a : {3, 4, 5, \\dots} \\to \\mathbb{R} \\), where the values \\( 3, 4, 5, \\dots \\) represent the indices of the sequence. Indeed, since the denominator becomes zero for \\( n = 2 \\), the term \\( a\_2 \\) is undefined. To avoid this singularity, we restrict the [domain](../determining-the-domain-of-a-function.md) to \\( n \\geq 3 \\). In this case, we write the sequence as:
\\[
(a\_n)\_{n \\geq 3} = \\left( \\frac{1}{n - 2} \\right)\_{n \\geq 3}
\\]
The first few terms of the sequence are:
\\[
a\_3 = 1, \\quad a\_4 = \\frac{1}{2}, \\quad a\_5 = \\frac{1}{3}, \\quad a\_6 = \\frac{1}{4}, \\quad a\_7 = \\frac{1}{5}, \\ \\dots
\\]
As we can see, this sequence decreases and converges to zero as \\( n \\to \\infty \\) (we will see later what this means).
## Recursively defined sequences
A recursive sequence is a sequence where each term is defined in terms of one or more of the preceding terms. To define such a sequence, two components are needed:
- An initial value.
- A recurrence relation, which determines how to compute each new term.
* * *
One of the most famous recursive sequences is the Fibonacci sequence, defined as:
\\[
\\begin{cases} a\_0 = 0, \\[0.5em] a\_1 = 1, \\[0.5em] a\_n = a\_{n-1} + a\_{n-2} \\quad \\text{for all } n \\geq 2 \\end{cases}
\\]
This means that every term is the sum of the two preceding ones. The first few terms of the sequence are:
\\[
\\begin{aligned} a\_0 &= 0 \\[0.5em] a\_1 &= 1 \\[0.5em] a\_2 &= 1 \\[0.5em] a\_3 &= 2 \\[0.5em] a\_4 &= 3 \\[0.5em] a\_5 &= 5 \\[0.5em] a\_6 &= 8 \\[0.5em] &\\vdots \\end{aligned}
\\]
##### Recursion is a common strategy in programming that allows complex tasks to be solved by repeatedly applying the same rule until a base case is reached. It’s especially effective for generating sequences and solving problems with a self-repeating structure.
## Monotonic sequences
A sequence can be classified based on how its terms evolve. In general, a sequence that satisfies any of these conditions is called a [monotonic sequence](../monotone-sequences.md):
- Constant: if every term is equal to the previous one: \\(a\_n = a\_{n+1} \\quad \\forall n \\in \\mathbb{N}\\).
- Increasing: if each term is greater than the previous one: \\(a\_n < a\_{n+1} \\quad \\forall n \\in \\mathbb{N}\\).
- Decreasing: if each term is less than the previous one: \\(a\_n > a\_{n+1} \\quad \\forall n \\in \\mathbb{N}\\).
- Non-decreasing: \\(a\_n \\leq a\_{n+1} \\quad \\forall n \\in \\mathbb{N}\\).
- Non-increasing: \\(a\_n \\geq a\_{n+1} \\quad \\forall n \\in \\mathbb{N}\\).
* * *
If a sequence \\( (a\_n)\_{n \\in \\mathbb{N}} \\) is monotonic, then it admits a limit and this limit is finite. Moreover, the following holds:
\\[
\\lim\_{n \\to +\\infty} a\_n = \\begin{cases} \\sup { a\_n : n \\in \\mathbb{N} } & \\text{if } (a\_n)\_{n \\in \\mathbb{N}} \\text{ is increasing} \\[0.5em] \\inf { a\_n : n \\in \\mathbb{N} } & \\text{if } (a\_n)\_{n \\in \\mathbb{N}} \\text{ is decreasing} \\end{cases}
\\]
This result guarantees that bounded monotonic sequences always converge, and their limit corresponds to the supremum or infimum depending on the direction of monotonicity.
## Glossary
- Sequence: an ordered collection of elements, each assigned to a specific position indexed by a natural number.
- Term: each individual element produced by a sequence.
- Index: a natural number that indicates the position of a term within a sequence.
- Monotonic sequence: a sequence that is either constant, increasing, decreasing, non-decreasing, or non-increasing.
- Limit: the value that the terms of a sequence approach as the index \\( n \\) goes to infinity.
- Supremum: the least upper bound of a set of numbers.
- Infimum: the greatest lower bound of a set of numbers.
