Sequences

What is a sequence

1.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 $a(n) \in \mathbb{R}$.

3.A sequence enumerates a countably infinite set when it is a bijection from $\mathbb{N}$ onto that set. Cardinality and countable sets relates this definition to finite enumerations and countability criteria.

4.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:

$$ a : \{n \in \mathbb{N} : n \geq n_0\} \to \mathbb{R}. $$

5.This means the sequence is defined for all natural numbers greater than or equal to some initial index $n_0$.

6.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:

$$ a_1 = 1, \quad a_2 = \frac{1}{2}, \quad \dots, \quad a_n = \frac{1}{n} \quad \forall n \in \mathbb{N}^+. $$

7.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

8.Consider, for example, the formula:

$$ a_n := \frac{1}{n - 2} $$

9.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:

$$ (a_n)_{n \geq 3} = \left( \frac{1}{n - 2} \right)_{n \geq 3} $$

10.The first few terms of the sequence are:

$$ a_3 = 1, \quad a_4 = \frac{1}{2}, \quad a_5 = \frac{1}{3}, \quad a_6 = \frac{1}{4}, \quad a_7 = \frac{1}{5}, \ \dots $$

11.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

12.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:

14.One of the most famous recursive sequences is the Fibonacci sequence, defined as:

$$ \begin{cases} a_0 = 0, \\[0.5em] a_1 = 1, \\[0.5em] a_n = a_{n-1} + a_{n-2} \quad \text{for all } n \geq 2 \end{cases} $$

15.This means that every term is the sum of the two preceding ones. The first few terms of the sequence are:

$$ \begin{aligned} a_0 &= 0 \\[0.5em] a_1 &= 1 \\[0.5em] a_2 &= 1 \\[0.5em] a_3 &= 2 \\[0.5em] a_4 &= 3 \\[0.5em] a_5 &= 5 \\[0.5em] a_6 &= 8 \\[0.5em] &\vdots \end{aligned} $$

16.

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

17.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:

19.Theorem 1. If a sequence $(a_n)_{n \in \mathbb{N}}$ is monotonic and bounded, then it admits a finite limit.

20.Theorem 2. If the sequence is monotonic but unbounded, then it diverges to $+\infty$ or to $-\infty$ depending on the direction of monotonicity. In the bounded case the limit is determined by the supremum or infimum of the range of the sequence:

$$ \lim_{n \to +\infty} a_n = \begin{cases} \sup \{ a_n : n \in \mathbb{N} \} & \text{if } (a_n)_{n \in \mathbb{N}} \text{ is increasing} \\[0.5em] \inf \{ a_n : n \in \mathbb{N} \} & \text{if } (a_n)_{n \in \mathbb{N}} \text{ is decreasing} \end{cases} $$

21.This result guarantees that bounded monotonic sequences always converge, and their limit corresponds to the supremum or infimum depending on the direction of monotonicity. The dedicated page on monotone sequences develops the proof of the theorem and treats the unbounded case in detail.

Cauchy sequences

22.A different way of describing the behaviour of a sequence focuses on the mutual distance between its terms rather than on the existence of a fixed limit. A sequence whose terms get arbitrarily close to one another from a certain index onward is called a Cauchy sequence.

23.In $\mathbb{R}$ every Cauchy sequence converges, and conversely every convergent sequence is a Cauchy sequence: this equivalence is the structural completeness of the real line, and it distinguishes $\mathbb{R}$ from the field of rational numbers.

Special sequences and further developments

24.Among the most important classes of sequences are the arithmetic sequences, whose terms differ by a constant amount, and the geometric sequences, whose terms are linked by a constant ratio. Their behaviour and the closed-form expression for the sum of the first $n$ terms are treated in the dedicated pages.

25.When the rule that generates each term yields a function rather than a number, the resulting object is a sequence of functions, whose convergence is studied in pointwise and uniform form.

26.The limit $e := \lim_{n \to \infty}(1 + 1/n)^n$ provides a celebrated example of a monotone bounded sequence and is discussed on the page about Euler's number. Many properties of sequences are most efficiently established by the principle of mathematical induction, which applies whenever a statement is parametrised by a natural number.

27.For sequences that do not converge but remain bounded, such as oscillating sequences, the ordinary limit does not exist. The superior and inferior limits extend the analysis to this case, identifying the largest and smallest cluster values around which the terms accumulate for arbitrarily large indices.