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, and can be manipulated with the usual algebraic operations on
series (addition, subtraction, multiplication, division,
partial sums, 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 languag ...
), 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 exampl ...
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 Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a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 con ...
, 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 a ...
, the method of
generating functions 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 called ...
s and
multisets, for instance allowing concise expressions for
recursively 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.
Rings of formal power series are
complete local rings, and this allows using
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
-like methods in the purely algebraic framework of
algebraic geometry 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. Prom ...
. 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 exampl ...
, 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 con ...
(or
Taylor series), one may think of a formal power series as a power series in which we ignore questions of
convergence 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 is 1. However, as a formal power series, we may ignore this completely; all that is relevant is the sequence of
coefficients
, −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
factorials
, 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):
:
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 manag ...
,
conditional
Conditional (if then) may refer to:
*Causal conditional, if X then Y, where X is a cause of Y
*Conditional probability, the probability of an event A given that another event B has occurred
*Conditional proof, in logic: a proof that asserts a co ...
and
uniform convergence 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 (3 ...
.
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