Convergent and Divergent Sequences
Behavior of a sequence
We introduced sequences as an ordered collection of elements, each assigned to a specific position indexed by a natural number. 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, 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 $ℓ \in \mathbb{R}$ if for every $\epsilon > 0$, there exists $n{0} \in \mathbb{N}$ such that:
\[| a_{n} - ℓ | < \epsilon \text{for all} n \geq n_{0} .\]In this case, we write:
\[\underset{n \rightarrow + \infty}{lim} a_{n} = ℓ \text{or} a_{n} \rightarrow ℓ \text{as} n \rightarrow + \infty .\]In other words, this means that the terms of the sequence get increasingly close to the number $ℓ$ as $n$ grows larger. No matter how tight a margin $\epsilon$, from a certain index onward all terms will stay within that distance from $ℓ$. For example, consider the following sequence: \(a_{n} = (( \frac{1}{n} ))_{n \geq} = ( 1 , \frac{1}{2} , \frac{1}{3} , \ldots )\)
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:
\[\underset{n \rightarrow + \infty}{lim} a_{n} = 0.\]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 $n \rightarrow + \infty$, using the formal definition of convergence.
To prove this, we must show that for every $\epsilon > 0$, there exists a natural number $n_{0}$ such that for all $n \geq n_{0}$:
\[| \frac{n}{n + 2} - 1 | < \epsilon\]Let’s simplify the absolute value expression:
\[| \frac{n}{n + 2} - 1 | = | \frac{- 2}{n + 2} | = \frac{2}{n + 2} .\]Now, we want:
\[\frac{2}{n + 2} < \epsilon\]Solving the inequality:
\[n + 2 > \frac{2}{\epsilon} \Rightarrow n > \frac{2}{\epsilon} - 2\]So we can define:
\[n_{0} = \lceil \frac{2}{\epsilon} - 2 \rceil\]From this point onward, every term of the sequence stays within a distance $\epsilon$ of the limit $1.$ Hence, by definition:
\[\underset{n \rightarrow + \infty}{lim} \frac{n}{n + 2} = 1.\]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 \(a_{n} > M \text{for all} n \geq n_{0}\) In this case, we write: \(\underset{n \rightarrow + \infty}{lim} a_{n} = + \infty \text{or} a_{n} \rightarrow + \infty \text{as} n \rightarrow + \infty\)
A sequence diverges to $- \infty$ if, for every $M < 0$, there exists an index $n_{0} \in \mathbb{N}$ such that \(a_{n} < M \text{for all} n \geq n_{0}\) In this case, we write: \(\underset{n \rightarrow + \infty}{lim} a_{n} = - \infty \text{or} a_{n} \rightarrow - \infty \text{as} n \rightarrow + \infty\)
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:
\[| a_{n} | \leq M \forall n \in \mathbb{N}\]We say that a sequence $a_{n}$ is bounded above if there exists a constant $M \in \mathbb{R}$ such that:
\[a_{n} \leq M \forall n \in \mathbb{N}\]We say that the sequence is bounded below if there exists a constant $M \in \mathbb{R}$ such that:
\[a_{n} \geq M \forall n \in \mathbb{N}\]Oscillating sequence
Oscillating sequences are a special type of bounded sequence. Let us consider the sequence:
\[( a_{n} )_{n \in \mathbb{N}} = ( ( - 1 )^{n} )_{n \in \mathbb{N}} = ( + 1 , - 1 , + 1 , - 1 , + 1 , - 1 , \ldots )\]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, 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 $\underset{n \rightarrow + \infty}{lim} a_{n} = a_{0} = 1.$
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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. As $n$ increases, each term doubles the previous one, leading to a rapid escalation in magnitude.