# Convergent and Divergent Sequences
Source: algebrica.org - CC BY-NC 4.0
https://algebrica.org/convergent-and-divergent-sequences/
Fetched from algebrica.org post 14468; source modified 2025-05-07T08:38:56.
## Behavior of a sequence
We introduced [sequences](../sequences.md) as an ordered collection of elements, each assigned to a specific position indexed by a [natural number](../types-of-numbers.md). To every sequence \\( (a\_n)\_{n \\in \\mathbb{N}} \\), there is an associated behavior of its terms \\( a\_n \\) that describes how they evolve as the index \\( n \\) increases. Analyzing this behavior helps determine whether the sequence converges to a finite [limit](../limits.md), diverges to infinity, or exhibits an oscillating pattern.
## Convergent sequence
A sequence \\( (a\_n)\_{n \\in \\mathbb{N}} \\) is said to be **convergent to the limit** \\( \\ell \\in \\mathbb{R} \\) if for every \\( \\varepsilon > 0 \\), there exists \\( n\_0 \\in \\mathbb{N} \\) such that:
\\[
|a\_n - \\ell| < \\varepsilon \\quad \\text{for all } n \\geq n\_0.
\\]
In this case, we write:
\\[
\\lim\_{n \\to +\\infty} a\_n = \\ell \\quad \\text{or} \\quad a\_n \\to \\ell \\quad \\text{as } n \\to +\\infty.
\\]
In other words, this means that the terms of the sequence get increasingly close to the number \\( \\ell \\) as \\( n \\) grows larger. No matter how tight a margin \\( \\varepsilon \\), from a certain index onward all terms will stay within that distance from \\( \\ell \\). For example, consider the following sequence:
\\[
a\_n = \\left( \\frac{1}{n}\\right)\_{n\\geq} = \\left(1, \\frac{1}{2}, \\frac{1}{3}, \\ldots \\right)
\\]
As \\( n \\) increases, the terms become smaller and smaller, approaching zero. This is a classic example of a sequence that converges to 0. A sequence is said to be **infinitesimal** when its terms get arbitrarily close to zero as the index grows and:
\\[
\\lim\_{n \\to +\\infty} a\_n = 0.
\\]
The limit of a sequence \\( (a\_n)\_{n \\in \\mathbb{N}} \\), if it exists, is unique.
## Example
Let’s consider the sequence defined by:
\\[
a\_n = \\frac{n}{n + 2}
\\]
We aim to demonstrate that this sequence converges to 1 as \\( n \\to +\\infty \\), using the formal definition of convergence.
* * *
To prove this, we must show that for every \\( \\varepsilon > 0 \\), there exists a natural number \\( n\_0 \\) such that for all \\( n \\geq n\_0 \\):
\\[
\\left| \\frac{n}{n + 2} - 1 \\right| < \\varepsilon
\\]
Let’s simplify the absolute value expression:
\\[
\\left| \\frac{n}{n + 2} - 1 \\right| = \\left| \\frac{-2}{n + 2} \\right| = \\frac{2}{n + 2}.
\\]
Now, we want:
\\[
\\frac{2}{n + 2} < \\varepsilon
\\]
Solving the inequality:
\\[
n + 2 > \\frac{2}{\\varepsilon} \\quad \\Rightarrow \\quad n > \\frac{2}{\\varepsilon} - 2
\\]
So we can define:
\\[
n\_0 = \\left\\lceil \\frac{2}{\\varepsilon} - 2 \\right\\rceil
\\]
From this point onward, every term of the sequence stays within a distance \\( \\varepsilon \\) of the limit \\(1.\\) Hence, by definition:
\\[
\\lim\_{n \\to +\\infty} \\frac{n}{n + 2} = 1.
\\]
## Divergent sequence
A sequence \\( (a\_n)\_{n \\in \\mathbb{N}} \\) is said to be **divergent** if it does not converge to a finite limit. This can happen in the following ways.
A sequence diverges to \\( +\\infty \\) if, for every \\( M > 0 \\), there exists an index \\( n\_0 \\in \\mathbb{N} \\) such that
\\[
a\_n > M \\quad \\text{for all } n \\geq n\_0
\\]
In this case, we write:
\\[
\\lim\_{n \\to +\\infty} a\_n = +\\infty \\quad \\text{or} \\quad a\_n \\to +\\infty \\text{ as } n \\to +\\infty
\\]
* * *
A sequence diverges to \\( -\\infty \\) if, for every \\( M < 0 \\), there exists an index \\( n\_0 \\in \\mathbb{N} \\) such that
\\[
a\_n < M \\quad \\text{for all } n \\geq n\_0
\\]
In this case, we write:
\\[
\\lim\_{n \\to +\\infty} a\_n = -\\infty \\quad \\text{or} \\quad a\_n \\to -\\infty \\text{ as } n \\to +\\infty
\\]
## Bounded sequence
A bounded sequence is a sequence of numbers whose terms always stay within a fixed, finite interval, no matter how large the index becomes. In formal terms, let \\( {a\_n} \\) be a sequence. We say that the sequence is bounded if there exists a constant \\( M > 0 \\) such that:
\\[
|a\_n| \\leq M \\quad \\forall n \\in \\mathbb{N}
\\]
We say that a sequence \\( {a\_n} \\) is bounded above if there exists a constant \\( M \\in \\mathbb{R} \\) such that:
\\[
a\_n \\leq M \\quad \\forall n \\in \\mathbb{N}
\\]
We say that the sequence is bounded below if there exists a constant \\( M \\in \\mathbb{R} \\) such that:
\\[
a\_n \\geq M \\quad \\forall n \\in \\mathbb{N}
\\]
## Oscillating sequence
Oscillating sequences are a special type of bounded sequence. Let us consider the sequence:
\\[
(a\_n)\_{n \\in \\mathbb{N}} = ((-1)^n)\_{n \\in \\mathbb{N}} = (+1, -1, +1, -1, +1, -1, \\dots)
\\]
As the index \\(n\\) increases, the terms of the sequence alternate consistently between \\(+1\\) and \\(-1\\). This type of sequence does not approach any finite value and is called an **oscillating sequence**. It does not converge to a finite limit, nor does it diverge to \\( +\\infty \\) or \\( -\\infty \\), and its terms continue to fluctuate between different values
## Geometric sequence
Let us consider an example of a sequence, called a [geometric sequence](../geometric-sequence.md), which can display different behaviors depending on the fixed real number \\( q \\). In general, a numerical sequence is called a geometric progression when the ratio between each term and its previous one is constant. More precisely, a geometric sequence is defined as follows:
\\[
a\_n := q^n
\\]
It exhibits the following behavior:
- It diverges to \\( +\\infty \\) if \\( q > 1 \\).
- It is constant (that is, \\( a\_n = a\_0 \\) for every \\( n \\in \\mathbb{N} \\)) if \\( q = 1 \\), and thus \\(\\lim\_{n \\to +\\infty} a\_n = a\_0 = 1.\\)
- It is infinitesimal if \\( |q| < 1 \\), meaning the terms approach zero.
- It is oscillatory (irregular) if \\( q \\leq -1 \\), due to alternating signs and unbounded growth.
As shown in the graph, when \\( q = 2 \\), the values of the geometric sequence \\( a\_n = q^n \\) grow [exponentially](../exponential-function.md). As \\( n \\) increases, each term doubles the previous one, leading to a rapid escalation in magnitude.
##### Take a closer look at the difference between an [arithmetic progression](../arithmetic-sequence.md) and a geometric progression to better understand how their structures and growth patterns differ.
