Sequences - Essential Dictionary II

Mathematical Writing - Vivaldi Franco 2014

Sequences
Essential Dictionary II

Franco Vivaldi1

(1)

School of Mathematical Sciences, Queen Mary, University of London, London, UK

Franco Vivaldi

Email: f.vivaldi@qmul.ac.uk

We further develop our mathematical dictionary, introducing the words sequence, sum, and equation. This will enable us to describe mathematical expressions of respectable complexity. As in the previous chapter, the last section contains advanced terms which are not essential for the rest of this book.

3.1 Sequences

A sequence is an ordered list of objects, not necessarily distinct, called the terms (or the elements) of the sequence. The terms of a sequence are represented by a common symbol, and each term is identified by an integer subscript:

$$\begin{aligned} (a_1,a_2,\ldots ,\,a_n) \qquad \qquad \qquad (a_1,a_2,\ldots \,). \end{aligned}$$

(3.1)

Here the common symbol is $$a$$, and the integer values assumed by the subscript begin from $$1$$. The quantity $$a_1$$ reads “$${\textit{a}}$$ sub $${\textit{1}}$$”, etc. Subscripts may also begin from 0, or from anywhere. The expression on the left denotes a finite sequence, the one on the right suggests that the sequence is infinite.

The length of a sequence is the number of its elements. Two sequences are equal if they have the same length, and if the corresponding terms are equal. If $$k$$ is an unspecified integer, then $$a_k$$ is called the general term of the sequence.

For example, the sequence of primes

$$ (p_1,p_2,p_3,\ldots \,)=(2,3,5,\ldots ) $$

is infinite. The general term $$p_k$$ is the $$k$$th prime number.

There are other notations for sequences, displaying the general term alongside information about the subscript range:

$$\begin{aligned} (a_k)_{k=1}^n \qquad \qquad (a_k)_1^n \qquad \qquad (a_k)_{k=1}^\infty \qquad \qquad (a_k)_{k\geqslant 1} \qquad \qquad (a_k). \end{aligned}$$

(3.2)

There are also the doubly-infinite sequences, where the subscript runs through all the integers:

$$ (a_k)_{k=-\infty }^{\infty }\,=\, (\ldots ,\,a_{-1},\,a_0,\,a_1,\ldots ). $$

In Sect. 6.2 we shall discuss the usage of these notations.

The ellipsis is ubiquitous in sequence notation. When we use it, we must make sure that the missing terms are defined unambiguously. Thus the general term of the sequence of monomials

$$ (2x,2x^2,2x^3,\ldots ) $$

is clearly equal to $$2x^k$$. However, the expression

$$ (3,5,7,\ldots ) $$

is ambiguous, because there are several plausible alternatives for the identity of the omitted terms, such as ($$9,11,13,\ldots )$$ or ($$11,13,17,\ldots $$). In the former case, we resolve the ambiguity by displaying the general term:

$$ (3,5,\ldots , 2k+1,\ldots ). $$

In the latter, we need an accompanying sentence.

A sub-sequence of a sequence $$(a_k)$$ is any sequence obtained from $$(a_k)$$ by deleting terms. For instance, the primes that give remainder 1 upon division by 4 form a sub-sequence of the sequence of primes:

$$ (5,13,17,29,\ldots ). $$

Some types of finite sequences have a dedicated terminology. A two-element sequence is an (ordered) pair, and a three-element sequence a triple. Occasionally one sees the terms quadruple or quintuple (I wouldn’t go much beyond that), while an $$n$$-element sequence may be called an $${\textit{n}}$$ -uple. A finite sequence of numbers may be called a vector, in which case we speak of dimension rather than length.

An infinite sequence $$(a_1,a_2,\ldots \,)$$ represents a function defined over the natural numbers. If the elements of the sequence belong to a set $$A$$, then such a function is defined as

$$ a:\mathbb {N}\rightarrow A\qquad \quad k\mapsto a_k. $$

We see that in the expression $$a_k$$, the symbol $$a$$ is the function’s name, the subscript $$k$$ is an element of the domain, and $$a_k\in A$$ is the value $$a(k)$$ of the function at $$k$$. This interpretation clarifies the meaning of expressions such as $$a_{k^2}$$: it is the composition of two functions, much like $$\sin (x^2)$$.

Many familiar constructs involve sequences. Let $$A$$ be a set of numbers, and let $$(a_0,\ldots ,a_n)$$ be a finite sequence of elements of $$A$$ with $$a_n\not =0$$. A polynomial over $$A$$ in the indeterminate $$x$$ is an expression of the type

$$\begin{aligned} a_0+a_1x+a_2x^2+\cdots +a_nx^n. \end{aligned}$$

(3.3)

In this notation, repeated addition is represented by the raised ellipsis. The elements of the sequence are called the coefficients of the polynomial, and the integer $$n$$ is its degree. The coefficients $$a_0$$ and $$a_n$$ are called, respectively, the constant and the leading coefficient. Each addendum in a polynomial is called a monomial, and a polynomial with two terms is a binomial. A polynomial of degree two is said to be quadratic; then we have cubic, quartic, and quintic polynomials. The set of all polynomials over the set $$A$$ with indeterminate $$x$$ is denoted by $$A[x]$$. For example

$$ x^2-x-1\,\in \mathbb {Z}[x] \qquad \qquad \qquad \frac{1}{2}-y^3\,\in \mathbb {Q}[y]. $$

A rational function is the ratio of two polynomials:

$$\begin{aligned} \frac{a_0+a_1x+\cdots +a_nx^n}{b_0+b_1x+\cdots +b_mx^m}. \end{aligned}$$

(3.4)

Its degree is the largest of $$m$$ and $$n$$ (assuming that $$a_nb_m\not =0$$). The set of all rational functions with coefficients in a set $$A$$ and indeterminate $$x$$ is denoted by $$A(x)$$.

A multivariate polynomial is a polynomial in more than one indeterminate. (The term univariate is used to differentiate from multivariate.)

$$ x^2y^2-\frac{1}{2}x^4-xy^3\,\in \mathbb {Q}[x,y]. $$

The total degree of each monomial is the sum of the degrees of the indeterminates, and the degree of a polynomial is the largest total degree among the monomials with non-zero coefficient. A multivariate polynomial is homogeneous if all monomials have the same total degree. The expression above may be described as

A homogeneous quartic polynomial in two indeterminates with rational coefficients.

EXAMPLE. Explain what is a polynomial. [ $$\not \varepsilon \,$$ ]

A polynomial is a finite sum. Each term, called a monomial, is the product of a coefficient (typically, a real or complex number) and one or more indeterminates, each raised to some non-negative integer power.