In 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.).
Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, since the operations that can be applied are different.
Formal power series are widely used in combinatorics for representing sequences of integers as generating functions.
In this context, a recurrence relation between the elements of a sequence may often be interpreted as a differential equation that the generating function satisfies.
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.
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 X5 term is given by For this reason, one may multiply formal power series without worrying about the usual questions of absolute, conditional and uniform convergence which arise in dealing with power series in the setting of analysis.
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.
Other examples include Similarly, many other operations that are carried out on polynomials can be extended to the formal power series setting, as explained below.
But the general construction of a completion of a metric space is more involved than what is needed here, and would make formal power series seem more complicated than they are.
, even though the latter is not an expression formed by the operations of addition and multiplication defined above (from which only finite sums can be constructed).
Formally, the sequence of partial sums of some infinite summation converges if for every fixed power of
It is also obvious that the limit of the sequence of partial sums is equal to the left hand side.
For instance the rule for multiplication can be restated simply as since only finitely many terms on the right affect any fixed
The above topology is the finest topology for which always converges as a summation to the formal power series designated by the same expression, and it often suffices to give a meaning to infinite sums and products, or other kinds of limits that one wishes to use to designate particular formal power series.
This more permissive approach is not however the standard when considering formal power series, as it would lead to convergence considerations that are as subtle as they are in analysis, while the philosophy of formal power series is on the contrary to make convergence questions as trivial as they can possibly be.
one may form the composition where the coefficients cn are determined by "expanding out" the powers of f(X): Here the sum is extended over all (k, j) with
Formal power series can be used to solve recurrences occurring in number theory and combinatorics.
One can use formal power series to prove several relations familiar from analysis in a purely algebraic setting.
In mathematical analysis, every convergent power series defines a function with values in the real or complex numbers.
Formal power series over certain special rings can also be interpreted as functions, but one has to be careful with the domain and codomain.
are defined in a similar way to a formal power series, except that we also allow finitely many terms of negative degree.
This sum has only finitely many nonzero terms because of the assumed vanishing of coefficients at sufficiently negative indices.
of formal Laurent series may be endowed with the structure of a topological ring by introducing the metric
is such that a sequence of its elements converges only if for each monomial Xα the corresponding coefficient stabilizes.
does not converge with respect to any J-adic topology on R, but clearly for each monomial the corresponding coefficient stabilizes.
If S is a commutative associative algebra over R, if I is an ideal of S such that the I-adic topology on S is complete, and if x1, ..., xr are elements of I, then there is a unique map
These formal power series are used to model the behavior of weighted automata, in theoretical computer science, when the coefficients
; because of the condition that the indexing set be well-ordered the product is well-defined, and we of course assume that two elements which differ by zero are the same.
This theory is due to Hans Hahn, who also showed that one obtains subfields when the number of (non-zero) terms is bounded by some fixed infinite cardinality.