In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a formal series is an infinite sum that is considered independently from any notion of
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
, and can be manipulated with the usual algebraic operations on
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
(addition, subtraction, multiplication, division,
partial sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
s, etc.).
A formal power series is a special kind of formal series, whose terms are of the form
where
is the
th power of a variable
(
is a non-negative
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
), and
is called the coefficient. Hence, power series can be viewed as a generalization of
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the
are used only as position-holders for the coefficients, so that the coefficient of
is the fifth term in the sequence. In
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, the method of
generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary seri ...
s uses formal power series to represent numerical
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s and
multiset
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that e ...
s, for instance allowing concise expressions for
recursively
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...
defined sequences regardless of whether the recursion can be explicitly solved. More generally, formal power series can include series with any finite (or countable) number of variables, and with coefficients in an arbitrary
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
.
Rings of formal power series are
complete local ring In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing c ...
s, and this allows using
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
-like methods in the purely algebraic framework of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
. They are analogous in many ways to
-adic integers, which can be defined as formal series of the powers of .
Introduction
A formal power series can be loosely thought of as an object that is like a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
, but with infinitely many terms. Alternatively, for those familiar with
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
(or
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
), one may think of a formal power series as a power series in which we ignore questions of
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
by not assuming that the variable ''X'' denotes any numerical value (not even an unknown value). For example, consider the series
:
If we studied this as a power series, its properties would include, for example, that its
radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
is 1. However, as a formal power series, we may ignore this completely; all that is relevant is the sequence of
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
s
, −3, 5, −7, 9, −11, ... In other words, a formal power series is an object that just records a sequence of coefficients. It is perfectly acceptable to consider a formal power series with the
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
s
, 1, 2, 6, 24, 120, 720, 5040, ... as coefficients, even though the corresponding power series diverges for any nonzero value of ''X''.
Arithmetic on formal power series is carried out by simply pretending that the series are polynomials. For example, if
:
then we add ''A'' and ''B'' term by term:
:
We can multiply formal power series, again just by treating them as polynomials (see in particular
Cauchy product In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy.
Definitions
The Cauchy product may apply to infinit ...
):
:
Notice that each coefficient in the product ''AB'' only depends on a ''finite'' number of coefficients of ''A'' and ''B''. For example, the ''X''
5 term is given by
:
For this reason, one may multiply formal power series without worrying about the usual questions of
absolute Absolute may refer to:
Companies
* Absolute Entertainment, a video game publisher
* Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK
* Absolute Software Corporation, specializes in security and data risk manage ...
,
conditional and
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
which arise in dealing with power series in the setting of
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
.
Once we have defined multiplication for formal power series, we can define multiplicative inverses as follows. The multiplicative inverse of a formal power series ''A'' is a formal power series ''C'' such that ''AC'' = 1, provided that such a formal power series exists. It turns out that if ''A'' has a multiplicative inverse, it is unique, and we denote it by ''A''
−1. Now we can define division of formal power series by defining ''B''/''A'' to be the product ''BA''
−1, provided that the inverse of ''A'' exists. For example, one can use the definition of multiplication above to verify the familiar formula
:
An important operation on formal power series is coefficient extraction. In its most basic form, the coefficient extraction operator