Sequences

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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. 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 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 $a: \mathbb{N}^+ \to \mathbb{R}$ 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 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 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:

  • 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.