Convergent and Divergent Sequences

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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 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 $\ell \in \mathbb{R}$ 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)

Convergent sequence.

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

Oscillating sequence.

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 $\lim_{n \to +\infty} a_n = a_0 = 1.$
  • 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. 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 and a geometric progression to better understand how their structures and growth patterns differ.