formal power series


Laurent seriesMathworldPlanetmath

Formal power seriesMathworldPlanetmath allow one to employ much of the analytical machinery of power seriesMathworldPlanetmath in settings which don’t have natural notions of convergence. They are also useful in order to compactly describe sequencesMathworldPlanetmath and to find closed formulas for recursively described sequences; this is known as the method of generating functions and will be illustrated below.

We start with a commutative ring R. We want to define the ring of formal power series over R in the variable X, denoted by R[[X]]; each element of this ring can be written in a unique way as an infinite sum of the form n=0anXn, where the coefficients an are elements of R; any choice of coefficients an is allowed. R[[X]] is actually a topological ring so that these infinite sums are well-defined and convergentMathworldPlanetmathPlanetmath. The additionPlanetmathPlanetmath and multiplication of such sums follows the usual laws of power series.

Formal construction

Start with the set R of all infiniteMathworldPlanetmathPlanetmath sequences in R. Define addition of two such sequences by


and multiplication by


This turns R into a commutative ring with multiplicative identityPlanetmathPlanetmath (1,0,0,…). We identify the element a of R with the sequence (a,0,0,…) and define X:=(0,1,0,0,). Then every element of R of the form (a0,a1,a2,,aN,0,0,) can be written as the finite sum


In order to extend this equation to infinite series, we need a metric on R. We define d((an),(bn))=2-k, where k is the smallest natural numberMathworldPlanetmath such that akbk (if there is not such k, then the two sequences are equal and we define their distance to be zero). This is a metric which turns R into a topological ring, and the equation


can now be rigorously proven using the notion of convergence arising from d; in fact, any rearrangement of the series convergesPlanetmathPlanetmath to the same limit.

This topological ring is the ring of formal power series over R and is denoted by R[[X]].


R[[X]] is an associative algebra over R which contains the ring R[X] of polynomials over R; the polynomials correspond to the sequences which end in zeros.

The geometric series formulaMathworldPlanetmathPlanetmath is valid in R[[X]]:


An element anXn of R[[X]] is invertiblePlanetmathPlanetmath in R[[X]] if and only if its constant coefficient a0 is invertible in R (see invertible formal power series).  This implies that the Jacobson radicalMathworldPlanetmath of R[[X]] is the ideal generated by X and the Jacobson radical of R.

Several algebraicMathworldPlanetmath properties of R are inherited by R[[X]]:

The metric space (R[[X]],d) is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. The topologyMathworldPlanetmath on R[[X]] is equal to the product topology on R where R is equipped with the discrete topology. It follows from TychonoffPlanetmathPlanetmath’s theorem that R[[X]] is compactPlanetmathPlanetmath if and only if R is finite. The topology on R[[X]] can also be seen as the I-adic topology, where I=(X) is the ideal generated by X (whose elements are precisely the formal power series with zero constant coefficient).

If R=K is a field, we can consider the quotient field of the integral domain K[[X]]; it is denoted by K((X)) and called a (formal) power series field. It is a topological field whose elements are called formal Laurent series; they can be uniquely written in the form


where M is an integer which depends on the Laurent series f.

Formal power series as functions

In analysisMathworldPlanetmath, every convergent power series defines a function with values in the real or complex numbersMathworldPlanetmathPlanetmath. Formal power series can also be interpreted as functions, but one has to be careful with the domain and codomain. If f=anXn is an element of R[[X]], if S is a commutativePlanetmathPlanetmathPlanetmath associative algebra over R, if I an ideal in S such that the I-adic topology on S is complete, and if x is an element of I, then we can define


This latter series is guaranteed to converge in S given the above assumptionsPlanetmathPlanetmath. Furthermore, we have




(unlike in the case of bona fide functions, these formulas are not definitions but have to proved).

Since the topology on R[[X]] is the (X)-adic topology and R[[X]] is complete, we can in particular apply power series to other power series, provided that the argumentsMathworldPlanetmathPlanetmath don’t have constant coefficients: f(0), f(X2-X) and f((1-X)-1-1) are all well-defined for any formal power series fR[[X]].

With this formalism, we can give an explicit formula for the multiplicative inverse of a power series f whose constant coefficient a=f(0) is invertible in R:


Differentiating formal power series

If f=n=0anXnR[[X]], we define the formal derivative of f as


This operationMathworldPlanetmath is R-linear, obeys the product ruleMathworldPlanetmath


and the chain ruleMathworldPlanetmath:


(in case g(0)=0).

In a sense, all formal power series are Taylor seriesMathworldPlanetmath, because if f=anXn, then


(here k! denotes the element 1×(1+1)×(1+1+1)×R.

One can also define differentiationMathworldPlanetmath for formal Laurent series in a natural way, and then the quotient ruleMathworldPlanetmath, in addition to the rules listed above, will also be valid.

Power series in several variables

The fastest way to define the ring R[[X1,,Xr]] of formal power series over R in r variables starts with the ring S=R[X1,,Xr] of polynomials over R. Let I be the ideal in S generated by X1,,Xr, consider the I-adic topology on S, and form its completion. This results in a complete topological ring containing S which is denoted by R[[X1,,Xr]].

For 𝐧=(n1,,nr)r, we write 𝐗𝐧=X1n1Xrnr. Then every element of R[[X1,,Xr]] can be written in a unique was as a sum


where the sum extends over all 𝐧r. These sums converge for any choice of the coefficients a𝐧R and the order in which the summation is carried out does not matter.

If J is the ideal in R[[X1,,Xr]] generated by X1,,Xr (i.e. J consists of those power series with zero constant coefficient), then the topology on R[[X1,,Xr]] is the J-adic topology.

Since R[[X1]] is a commutative ring, we can define its power series ring, say R[[X1]][[X2]]. This ring is naturally isomorphicPlanetmathPlanetmathPlanetmath to the ring R[[X1,X2]] just defined, but as topological rings the two are different.

If K=R is a field, then K[[X1,,Xr]] is a unique factorization domainMathworldPlanetmath.

Similar to the situation described above, we can “apply” power series in several variables to other power series with zero constant coefficients. It is also possible to define partial derivativesMathworldPlanetmath for formal power series in a straightforward way. Partial derivatives commute, as they do for continuously differentiable functions.


One can use formal power series to prove several relationsMathworldPlanetmathPlanetmath familar from analysis in a purely algebraic setting. Consider for instance the following elements of [[X]]:


Then one can easily show that




as well as


(the latter being valid in the ring [[X,Y]]).

As an example of the method of generating functions, consider the problem of finding a closed formula for the Fibonacci numbersMathworldPlanetmath fn defined by fn+2=fn+1+fn, f0=0, and f1=1. We work in the ring [[X]] and define the power series


f is called the generating function for the sequence (fn). The generating function for the sequence (fn-1) is Xf while that for (fn-2) is X2f. From the recurrence relation, we therefore see that the power series Xf+X2f agrees with f except for the first two coefficients. Taking these into account, we find that


(this is the crucial step; recurrence relations can almost always be translated into equations for the generating functions). Solving this equation for f, we get


Using the golden ratioMathworldPlanetmath ϕ1=(1+5)/2 and ϕ2=(1-5)/2, we can write the latter expression as


These two power series are known explicitly because they are geometric series; comparing coefficients, we find the explicit formula


In algebra, the ring K[[X1,,Xr]] (where K is a field) is often used as the “standard, most general” complete local ring over K.

Universal property

The power series ring R[[X1,,Xr]] can be characterized by the following universal property: if S is a commutative associative algebra over R, if I is an ideal in S such that the I-adic topology on S is complete, and if x1,,xrI are given, then there exists a unique Φ:R[[X1,,Xr]]S with the following properties:

  • Φ is an R-algebra homomorphism

  • Φ is continuousMathworldPlanetmathPlanetmath (

  • Φ(Xi)=xi for i=1,,r.

Title formal power series
Canonical name FormalPowerSeries
Date of creation 2013-03-22 12:49:30
Last modified on 2013-03-22 12:49:30
Owner AxelBoldt (56)
Last modified by AxelBoldt (56)
Numerical id 14
Author AxelBoldt (56)
Entry type Topic
Classification msc 13H05
Classification msc 13B35
Classification msc 13J05
Classification msc 13F25
Related topic PowerSeries
Related topic SumOfKthPowersOfTheFirstNPositiveIntegers
Related topic PolynomialRingOverIntegralDomain
Related topic FiniteRingHasNoProperOverrings
Defines formal power series
Defines generating function
Defines formal Laurent series
Defines power series field