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Inmathematics, aformal series is an infinite sum that is considered independently from any notion ofconvergence, and can be manipulated with the usual algebraic operations onseries (addition, subtraction, multiplication, division,partial sums, etc.).

Aformal power series is a special kind of formal series, of the form$${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n=a_0+a_{1}x+a_2{x}^2+\cdots ,}$$where the${\displaystyle a_n,}$ calledcoefficients, are numbers or, more generally, elements of somering, and the${\displaystyle x^n}$ are formal powers of the symbol${\displaystyle x}$ that is called anindeterminate or, commonly, avariable. Hence, power series can be viewed as a generalization ofpolynomials where the number of terms is allowed to be infinite, and differ from usualpower series by the absence of convergence requirements, which implies that a power series may not represent a function of its variable. Formal power series are inone to one correspondence with theirsequences of coefficients, but the two concepts must not be confused, since the operations that can be applied are different.

A formal power series with coefficients in a ring${\displaystyle R}$ is called a formal power series over${\displaystyle R.}$ The formal power series over a ring${\displaystyle R}$ form a ring, commonly denoted by${\displaystyle R[[x]].}$ (It can be seen as the(x)-adic completion of thepolynomial ring${\displaystyle R[x],}$ in the same way as thep-adic integers are thep-adic completion of the ring of the integers.)

Formal powers series in several indeterminates are defined similarly by replacing the powers of a single indeterminate bymonomials in several indeterminates.

Formal power series are widely used incombinatorics for representing sequences of integers asgenerating functions. In this context, arecurrence relation between the elements of a sequence may often be interpreted as adifferential equation that the generating function satisfies. This allows using methods ofcomplex analysis for combinatorial problems (seeanalytic combinatorics).

A formal power series can be loosely thought of as an object that is like apolynomial, but with infinitely many terms. Alternatively, for those familiar withpower series (orTaylor series), one may think of a formal power series as a power series in which we ignore questions ofconvergence by not assuming that the variableX denotes any numerical value (not even an unknown value). For example, consider the series$${\displaystyle A=1-3X+5X^2-7X^3+9X^4-11X^5+\cdots .}$$If we studied this as a power series, its properties would include, for example, that itsradius of convergence is 1 by theCauchy–Hadamard theorem. However, as a formal power series, we may ignore this completely; all that is relevant is the sequence ofcoefficients [1, −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 thefactorials [1, 1, 2, 6, 24, 120, 720, 5040, ... ] as coefficients, even though the corresponding power series diverges for any nonzero value ofX.

Algebra on formal power series is carried out by simply pretending that the series are polynomials. For example, if

${\displaystyle B=2X+4X^3+6X^5+\cdots ,}$

then we addA andB term by term:

${\displaystyle A+B=1-X+5X^2-3X^3+9X^4-5X^5+\cdots .}$

We can multiply formal power series, again just by treating them as polynomials (see in particularCauchy product):

${\displaystyle AB=2X-6X^2+14X^3-26X^4+44X^5+\cdots .}$

Notice that each coefficient in the productAB only depends on afinite number of coefficients ofA andB. For example, theX⁵ term is given by

${\displaystyle 44X^5=(1\times 6X^5)+(5X^2\times 4X^3)+(9X^4\times 2X).}$

For this reason, one may multiply formal power series without worrying about the usual questions ofabsolute,conditional anduniform convergence which arise in dealing with power series in the setting ofanalysis.

Once we have defined multiplication for formal power series, we can define multiplicative inverses as follows. The multiplicative inverse of a formal power seriesA is a formal power seriesC such thatAC = 1, provided that such a formal power series exists. It turns out that ifA has a multiplicative inverse, it is unique, and we denote it byA⁻¹. Now we can define division of formal power series by definingB/A to be the productBA⁻¹, provided that the inverse ofA exists. For example, one can use the definition of multiplication above to verify the familiar formula

${\displaystyle \frac{1}{1+X}=1-X+X^2-X^3+X^4-X^5+\cdots .}$

An important operation on formal power series is coefficient extraction. In its most basic form, the coefficient extraction operator${\displaystyle [X^n]}$ applied to a formal power series${\displaystyle A}$ in one variable extracts the coefficient of the${\displaystyle n}$th power of the variable, so that${\displaystyle [X^2]A=5}$ and${\displaystyle [X^5]A=-11}$. 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.

If one considers the set of all formal power series inX with coefficients in acommutative ringR, the elements of this set collectively constitute another ring which is written${\displaystyle R[[X]],}$ and called thering of formal power series in the variable X overR.

One can characterize${\displaystyle R[[X]]}$ abstractly as thecompletion of thepolynomial ring${\displaystyle R[X]}$ equipped with a particularmetric. This automatically gives${\displaystyle R[[X]]}$ the structure of atopological ring (and even of a complete metric space). 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. It is possible to describe${\displaystyle R[[X]]}$ more explicitly, and define the ring structure and topological structure separately, as follows.

As a set,${\displaystyle R[[X]]}$ can be constructed as the set${\displaystyle R^{\mathbb{N}}}$ of all infinite sequences of elements of${\displaystyle R}$, indexed by thenatural numbers (taken to include 0). Designating a sequence whose term at index${\displaystyle n}$ is${\displaystyle a_n}$ by${\displaystyle (a_n)}$, one defines addition of two such sequences by

${\displaystyle (a_n{)}_{n\in \mathbb{N}}+(b_n{)}_{n\in \mathbb{N}}={\left(a_n+b_n\right)}_{n\in \mathbb{N}}}$

and multiplication by

${\displaystyle (a_n{)}_{n\in \mathbb{N}}\times (b_n{)}_{n\in \mathbb{N}}={\left(\sum _{k=0}^n{a}_k{b}_{n-k}\right)}_{n\in \mathbb{N}}.}$

This type of product is called theCauchy product of the two sequences of coefficients, and is a sort of discreteconvolution. With these operations,${\displaystyle R^{\mathbb{N}}}$ becomes a commutative ring with zero element${\displaystyle (0,0,0,\dots )}$ and multiplicative identity${\displaystyle (1,0,0,\dots )}$.

The product is in fact the same one used to define the product of polynomials in one indeterminate, which suggests using a similar notation. One embeds${\displaystyle R}$ into${\displaystyle R[[X]]}$ by sending any (constant)${\displaystyle a\in R}$ to the sequence${\displaystyle (a,0,0,\dots )}$ and designates the sequence${\displaystyle (0,1,0,0,\dots )}$ by${\displaystyle X}$; then using the above definitions every sequence with only finitely many nonzero terms can be expressed in terms of these special elements as

${\displaystyle (a_0,a_1,a_2,\dots ,a_n,0,0,\dots )=a_0+a_{1}X+\cdots +a_n{X}^n=\sum _{i=0}^n{a}_i{X}^i;}$

these are precisely the polynomials in${\displaystyle X}$. Given this, it is quite natural and convenient to designate a general sequence${\displaystyle (a_n{)}_{n\in \mathbb{N}}}$ by the formal expression${\displaystyle \sum _{i\in \mathbb{N}}{a}_i{X}^i}$, even though the latteris not an expression formed by the operations of addition and multiplication defined above (from which only finite sums can be constructed). This notational convention allows reformulation of the above definitions as

${\displaystyle \left(\sum _{i\in \mathbb{N}}{a}_i{X}^i\right)+\left(\sum _{i\in \mathbb{N}}{b}_i{X}^i\right)=\sum _{i\in \mathbb{N}}(a_i+b_i)X^i}$

and

${\displaystyle \left(\sum _{i\in \mathbb{N}}{a}_i{X}^i\right)\times \left(\sum _{i\in \mathbb{N}}{b}_i{X}^i\right)=\sum _{n\in \mathbb{N}}\left(\sum _{k=0}^n{a}_k{b}_{n-k}\right)X^n.}$

which is quite convenient, but one must be aware of the distinction between formal summation (a mere convention) and actual addition.

Having stipulated conventionally that

${\displaystyle (a_0,a_1,a_2,a_3,\dots )=\sum _{i=0}^{\mathrm{\infty }}{a}_i{X}^i,}$

1

one would like to interpret the right hand side as a well-defined infinite summation. To that end, a notion of convergence in${\displaystyle R^{\mathbb{N}}}$ is defined and atopology on${\displaystyle R^{\mathbb{N}}}$ is constructed. There are several equivalent ways to define the desired topology.

• We may give${\displaystyle R^{\mathbb{N}}}$ theproduct topology, where each copy of${\displaystyle R}$ is given thediscrete topology.
• We may give${\displaystyle R^{\mathbb{N}}}$ theI-adic topology, where${\displaystyle I=(X)}$ is the ideal generated by${\displaystyle X}$, which consists of all sequences whose first term${\displaystyle a_0}$ is zero.
• The desired topology could also be derived from the followingmetric. The distance between distinct sequences${\displaystyle (a_n),(b_n)\in R^{\mathbb{N}},}$ is defined to be$${\displaystyle d((a_n),(b_n))=2^{-k},}$$ where${\displaystyle k}$ is the smallestnatural number such that${\displaystyle a_k\ne b_k}$; the distance between two equal sequences is of course zero.

Informally, two sequences${\displaystyle (a_n)}$ and${\displaystyle (b_n)}$ become closer and closer if and only if more and more of their terms agree exactly. Formally, the sequence ofpartial sums of some infinite summation converges if for every fixed power of${\displaystyle X}$ the coefficient stabilizes: there is a point beyond which all further partial sums have the same coefficient. This is clearly the case for the right hand side of (1), regardless of the values${\displaystyle a_n}$, since inclusion of the term for${\displaystyle i=n}$ gives the last (and in fact only) change to the coefficient of${\displaystyle X^n}$. It is also obvious that thelimit of the sequence of partial sums is equal to the left hand side.

This topological structure, together with the ring operations described above, form a topological ring. This is called thering of formal power series over${\displaystyle R}$ and is denoted by${\displaystyle R[[X]]}$. The topology has the useful property that an infinite summation converges if and only if the sequence of its terms converges to 0, which just means that any fixed power of${\displaystyle X}$ occurs in only finitely many terms.

The topological structure allows much more flexible usage of infinite summations. For instance the rule for multiplication can be restated simply as

${\displaystyle \left(\sum _{i\in \mathbb{N}}{a}_i{X}^i\right)\times \left(\sum _{i\in \mathbb{N}}{b}_i{X}^i\right)=\sum _{i,j\in \mathbb{N}}{a}_i{b}_j{X}^{i+j},}$

since only finitely many terms on the right affect any fixed${\displaystyle X^n}$. Infinite products are also defined by the topological structure; it can be seen that an infinite product converges if and only if the sequence of its factors converges to 1 (in which case the product is nonzero) or infinitely many factors have no constant term (in which case the product is zero).

The above topology is thefinest topology for which

${\displaystyle \sum _{i=0}^{\mathrm{\infty }}{a}_i{X}^i}$

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. It can however happen occasionally that one wishes to use a coarser topology, so that certain expressions become convergent that would otherwise diverge. This applies in particular when the base ring${\displaystyle R}$ already comes with a topology other than the discrete one, for instance if it is also a ring of formal power series.

In the ring of formal power series${\displaystyle \mathbb{Z}[[X]][[Y]]}$, the topology of above construction only relates to the indeterminate${\displaystyle Y}$, since the topology that was put on${\displaystyle \mathbb{Z}[[X]]}$ has been replaced by the discrete topology when defining the topology of the whole ring. So

${\displaystyle \sum _{i=0}^{\mathrm{\infty }}X{Y}^i}$

converges (and its sum can be written as${\displaystyle \frac{X}{1-Y}}$); however

${\displaystyle \sum _{i=0}^{\mathrm{\infty }}{X}^{i}Y}$

would be considered to be divergent, since every term affects the coefficient of${\displaystyle Y}$. This asymmetry disappears if the power series ring in${\displaystyle Y}$ is given the product topology where each copy of${\displaystyle \mathbb{Z}[[X]]}$ is given its topology as a ring of formal power series rather than the discrete topology. With this topology, a sequence of elements of${\displaystyle \mathbb{Z}[[X]][[Y]]}$ converges if the coefficient of each power of${\displaystyle Y}$ converges to a formal power series in${\displaystyle X}$, a weaker condition than stabilizing entirely. For instance, with this topology, in the second example given above, the coefficient of${\displaystyle Y}$converges to${\displaystyle \frac{1}{1-X}}$, so the whole summation converges to${\displaystyle \frac{Y}{1-X}}$.

This way of defining the topology is in fact the standard one for repeated constructions of rings of formal power series, and gives the same topology as one would get by taking formal power series in all indeterminates at once. In the above example that would mean constructing${\displaystyle \mathbb{Z}[[X,Y]]}$ and here a sequence converges if and only if the coefficient of every monomial${\displaystyle X^i{Y}^j}$ stabilizes. This topology, which is also the${\displaystyle I}$-adic topology, where${\displaystyle I=(X,Y)}$ is the ideal generated by${\displaystyle X}$ and${\displaystyle Y}$, still enjoys the property that a summation converges if and only if its terms tend to 0.

The same principle could be used to make other divergent limits converge. For instance in${\displaystyle \mathbb{R}[[X]]}$ the limit

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{X}{n}\right)}^n}$

does not exist, so in particular it does not converge to

${\displaystyle \exp (X)=\sum _{n\in \mathbb{N}}\frac{X^n}{n!}.}$

This is because for${\displaystyle i\ge 2}$ the coefficient${\displaystyle (\genfrac{}{}{0ex}{}{n}{i})/n^i}$ of${\displaystyle X^i}$ does not stabilize as${\displaystyle n\to \mathrm{\infty }}$. It does however converge in the usual topology of${\displaystyle \mathbb{R}}$, and in fact to the coefficient${\displaystyle \frac{1}{i!}}$ of${\displaystyle \exp (X)}$. Therefore, if one would give${\displaystyle \mathbb{R}[[X]]}$ the product topology of${\displaystyle {\mathbb{R}}^{\mathbb{N}}}$ where the topology of${\displaystyle \mathbb{R}}$ is the usual topology rather than the discrete one, then the above limit would converge to${\displaystyle \exp (X)}$. 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 inanalysis, while the philosophy of formal power series is on the contrary to make convergence questions as trivial as they can possibly be. With this topology it wouldnot be the case that a summation converges if and only if its terms tend to 0.

The ring${\displaystyle R[[X]]}$ may be characterized by the followinguniversal property. If${\displaystyle S}$ is a commutative associative algebra over${\displaystyle R}$, if${\displaystyle I}$ is an ideal of${\displaystyle S}$ such that the${\displaystyle I}$-adic topology on${\displaystyle S}$ is complete, and if${\displaystyle x}$ is an element of${\displaystyle I}$, then there is aunique${\displaystyle \mathrm{\Phi }:R[[X]]\to S}$ with the following properties:

• ${\displaystyle \mathrm{\Phi }}$ is an${\displaystyle R}$-algebra homomorphism
• ${\displaystyle \mathrm{\Phi }}$ is continuous
• ${\displaystyle \mathrm{\Phi }(X)=x}$.

One can perform algebraic operations on power series to generate new power series.^[1][2]

For anynatural numbern, thenth power of a formal power seriesS is defined recursively byIfa₀ is invertible in the ring of coefficients, one can prove that in the expansion$${\displaystyle (\sum _{k=0}^{\mathrm{\infty }}{a}_k{X}^k{)}^n=\sum _{m=0}^{\mathrm{\infty }}{c}_m{X}^m,}$$the coefficients are given by${\displaystyle c_0=a_0^n}$ and$${\displaystyle c_m=\frac{1}{m{a}_0}\sum _{k=1}^m(kn-m+k)a_k{c}_{m-k}}$$for${\displaystyle m\ge 1}$ ifm is invertible in the ring of coefficients.^[3][4][5][a]In the case of formal power series with complex coefficients, its complex powers are well defined for seriesf with constant term equal to1. In this case,${\displaystyle f^{\alpha }}$ can be defined either by composition with thebinomial series(1 +x)^α, or by composition with the exponential and the logarithmic series,${\displaystyle f^{\alpha }=\exp (\alpha \log (f)),}$ or as the solution of the differential equation (in terms of series)${\displaystyle f(f^{\alpha }{)}^{\prime }=\alpha f^{\alpha }{f}^{\prime }}$ with constant term1; the three definitions are equivalent. Theexponent rules${\displaystyle (f^{\alpha }{)}^{\beta }=f^{\alpha \beta }}$ and${\displaystyle f^{\alpha }{g}^{\alpha }=(fg{)}^{\alpha }}$ easily follow for formal power seriesf,g.

The series

${\displaystyle A=\sum _{n=0}^{\mathrm{\infty }}{a}_n{X}^n\in R[[X]]}$

is invertible in${\displaystyle R[[X]]}$ if and only if its constant coefficient${\displaystyle a_0}$ is invertible in${\displaystyle R}$. This condition is necessary, for the following reason: if we suppose that${\displaystyle A}$ has an inverse${\displaystyle B=b_0+b_{1}x+\cdots }$ then theconstant term${\displaystyle a_0{b}_0}$ of${\displaystyle A\cdot B}$ is the constant term of the identity series, i.e. it is 1. This condition is also sufficient; we may compute the coefficients of the inverse series${\displaystyle B}$ via the explicit recursive formula

An important special case is that thegeometric series formula is valid in${\displaystyle R[[X]]}$:

${\displaystyle (1-X{)}^{-1}=\sum _{n=0}^{\mathrm{\infty }}{X}^n.}$

If${\displaystyle R=K}$ is a field, then a series is invertible if and only if the constant term is non-zero, i.e. if and only if the series is not divisible by${\displaystyle X}$. This means that${\displaystyle K[[X]]}$ is adiscrete valuation ring with uniformizing parameter${\displaystyle X}$.

The computation of a quotient${\displaystyle f/g=h}$

${\displaystyle \frac{\sum _{n=0}^{\mathrm{\infty }}{b}_n{X}^n}{\sum _{n=0}^{\mathrm{\infty }}{a}_n{X}^n}=\sum _{n=0}^{\mathrm{\infty }}{c}_n{X}^n,}$

assuming the denominator is invertible (that is,${\displaystyle a_0}$ is invertible in the ring of scalars), can be performed as a product${\displaystyle f}$ and the inverse of${\displaystyle g}$, or directly equating the coefficients in${\displaystyle f=gh}$:

${\displaystyle c_n=\frac{1}{a_0}\left(b_n-\sum _{k=1}^n{a}_k{c}_{n-k}\right).}$

The coefficient extraction operator applied to a formal power series

${\displaystyle f(X)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{X}^n}$

inX is written

${\displaystyle \left[X^m\right]f(X)}$

and extracts the coefficient ofX^m, so that

${\displaystyle \left[X^m\right]f(X)=\left[X^m\right]\sum _{n=0}^{\mathrm{\infty }}{a}_n{X}^n=a_m.}$

Given two formal power series

${\displaystyle f(X)=\sum _{n=1}^{\mathrm{\infty }}{a}_n{X}^n=a_{1}X+a_2{X}^2+\cdots }$

${\displaystyle g(X)=\sum _{n=0}^{\mathrm{\infty }}{b}_n{X}^n=b_0+b_{1}X+b_2{X}^2+\cdots }$

such that${\displaystyle a_0=0,}$one may form thecomposition

${\displaystyle g(f(X))=\sum _{n=0}^{\mathrm{\infty }}{b}_n(f(X){)}^n=\sum _{n=0}^{\mathrm{\infty }}{c}_n{X}^n,}$

where the coefficientsc_n are determined by "expanding out" the powers off(X):

${\displaystyle c_n:=\sum _{k\in \mathbb{N},|j|=n}{b}_k{a}_{j_1}{a}_{j_2}\cdots a_{j_k}.}$

Here the sum is extended over all (k,j) with${\displaystyle k\in \mathbb{N}}$ and${\displaystyle j\in {\mathbb{N}}_{+}^k}$ with${\displaystyle |j|:=j_1+\cdots +j_k=n.}$

Since${\displaystyle a_0=0,}$ one must have${\displaystyle k\le n}$ and${\displaystyle j_i\le n}$ for every${\displaystyle i.}$ This implies that the above sum is finite and that the coefficient${\displaystyle c_n}$ is the coefficient of${\displaystyle X^n}$ in the polynomial${\displaystyle g_n(f_n(X))}$, where${\displaystyle f_n}$ and${\displaystyle g_n}$ are the polynomials obtained by truncating the series at${\displaystyle x^n,}$ that is, by removing all terms involving a power of${\displaystyle X}$ higher than${\displaystyle n.}$

A more explicit description of these coefficients is provided byFaà di Bruno's formula, at least in the case where the coefficient ring is a field ofcharacteristic 0.

Composition is only valid when${\displaystyle f(X)}$ hasno constant term, so that each${\displaystyle c_n}$ depends on only a finite number of coefficients of${\displaystyle f(X)}$ and${\displaystyle g(X)}$. In other words, the series for${\displaystyle g(f(X))}$ converges in thetopology of${\displaystyle R[[X]]}$.

Assume that the ring${\displaystyle R}$ has characteristic 0 and the nonzero integers are invertible in${\displaystyle R}$. If one denotes by${\displaystyle \exp (X)}$ the formal power series

${\displaystyle \exp (X)=1+X+\frac{X^2}{2!}+\frac{X^3}{3!}+\frac{X^4}{4!}+\cdots ,}$

then the equality

${\displaystyle \exp (\exp (X)-1)=1+X+X^2+\frac{5X^3}{6}+\frac{5X^4}{8}+\cdots }$

makes perfect sense as a formal power series, since the constant coefficient of${\displaystyle \exp (X)-1}$ is zero.

Whenever a formal series

${\displaystyle f(X)=\sum _k{f}_k{X}^k\in R[[X]]}$

hasf₀ = 0 andf₁ being an invertible element ofR, there exists a series

${\displaystyle g(X)=\sum _k{g}_k{X}^k}$

that is thecomposition inverse of${\displaystyle f}$, meaning that composing${\displaystyle f}$ with${\displaystyle g}$ gives the series representing theidentity function${\displaystyle x=0+1x+0x^2+0x^3+\cdots }$. The coefficients of${\displaystyle g}$ may be found recursively by using the above formula for the coefficients of a composition, equating them with those of the composition identityX (that is 1 at degree 1 and 0 at every degree greater than 1). In the case when the coefficient ring is a field of characteristic 0, theLagrange inversion formula (discussed below) provides a powerful tool to compute the coefficients ofg, as well as the coefficients of the (multiplicative) powers ofg.

Given a formal power series

${\displaystyle f=\sum _{n\ge 0}{a}_n{X}^n\in R[[X]],}$

we define itsformal derivative, denotedDf orf ′, by

${\displaystyle Df=f^{\prime }=\sum _{n\ge 1}{a}_{n}n{X}^{n-1}.}$

The symbolD is called theformal differentiation operator. This definition simply mimics term-by-term differentiation of a polynomial.

This operation isR-linear:

${\displaystyle D(af+bg)=a\cdot Df+b\cdot Dg}$

for anya,b inR and anyf,g in${\displaystyle R[[X]].}$ Additionally, the formal derivative has many of the properties of the usualderivative of calculus. For example, theproduct rule is valid:

${\displaystyle D(fg)=f\cdot (Dg)+(Df)\cdot g,}$

and thechain rule works as well:

${\displaystyle D(f\circ g)=(Df\circ g)\cdot Dg,}$

whenever the appropriate compositions of series are defined (see above undercomposition of series).

Thus, in these respects formal power series behave likeTaylor series. Indeed, for thef defined above, we find that

${\displaystyle (D^{k}f)(0)=k!a_k,}$

whereD^k denotes thekth formal derivative (that is, the result of formally differentiatingk times).

If${\displaystyle R}$ is a ring with characteristic zero and the nonzero integers are invertible in${\displaystyle R}$, then given a formal power series

${\displaystyle f=\sum _{n\ge 0}{a}_n{X}^n\in R[[X]],}$

we define itsformal antiderivative orformal indefinite integral by

${\displaystyle D^{-1}f=\int fdX=C+\sum _{n\ge 0}{a}_n\frac{X^{n+1}}{n+1}.}$

for any constant${\displaystyle C\in R}$.

This operation isR-linear:

${\displaystyle D^{-1}(af+bg)=a\cdot D^{-1}f+b\cdot D^{-1}g}$

for anya,b inR and anyf,g in${\displaystyle R[[X]].}$ Additionally, the formal antiderivative has many of the properties of the usualantiderivative of calculus. For example, the formal antiderivative is theright inverse of the formal derivative:

${\displaystyle D(D^{-1}(f))=f}$

for any${\displaystyle f\in R[[X]]}$.

${\displaystyle R[[X]]}$ is anassociative algebra over${\displaystyle R}$ which contains the ring${\displaystyle R[X]}$ of polynomials over${\displaystyle R}$; the polynomials correspond to the sequences which end in zeros.

TheJacobson radical of${\displaystyle R[[X]]}$ is theideal generated by${\displaystyle X}$ and the Jacobson radical of${\displaystyle R}$; this is implied by the element invertibility criterion discussed above.

Themaximal ideals of${\displaystyle R[[X]]}$ all arise from those in${\displaystyle R}$ in the following manner: an ideal${\displaystyle M}$ of${\displaystyle R[[X]]}$ is maximal if and only if${\displaystyle M\cap R}$ is a maximal ideal of${\displaystyle R}$ and${\displaystyle M}$ is generated as an ideal by${\displaystyle X}$ and${\displaystyle M\cap R}$.

Several algebraic properties of${\displaystyle R}$ are inherited by${\displaystyle R[[X]]}$:

• if${\displaystyle R}$ is alocal ring, then so is${\displaystyle R[[X]]}$ (with the set ofnon units the unique maximal ideal),
• if${\displaystyle R}$ isNoetherian, then so is${\displaystyle R[[X]]}$ (a version of theHilbert basis theorem),
• if${\displaystyle R}$ is anintegral domain, then so is${\displaystyle R[[X]]}$, and
• if${\displaystyle K}$ is afield, then${\displaystyle K[[X]]}$ is adiscrete valuation ring.

The metric space${\displaystyle (R[[X]],d)}$ iscomplete.

The ring${\displaystyle R[[X]]}$ iscompact if and only ifR isfinite. This follows fromTychonoff's theorem and the characterisation of the topology on${\displaystyle R[[X]]}$ as a product topology.

The ring of formal power series with coefficients in acomplete local ring satisfies theWeierstrass preparation theorem.

Formal power series can be used to solve recurrences occurring in number theory and combinatorics. For an example involving finding a closed form expression for theFibonacci numbers, see the article onExamples of generating functions.

One can use formal power series to prove several relations familiar from analysis in a purely algebraic setting. Consider for instance the following elements of${\displaystyle \mathbb{Q}[[X]]}$:

${\displaystyle \sin (X):=\sum _{n\ge 0}\frac{(-1{)}^n}{(2n+1)!}{X}^{2n+1}}$

${\displaystyle \cos (X):=\sum _{n\ge 0}\frac{(-1{)}^n}{(2n)!}{X}^{2n}}$

Then one can show that

${\displaystyle {\sin }^2(X)+{\cos }^2(X)=1,}$

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }X}\sin (X)=\cos (X),}$

${\displaystyle \sin (X+Y)=\sin (X)\cos (Y)+\cos (X)\sin (Y).}$

The last one being valid in the ring${\displaystyle \mathbb{Q}[[X,Y]].}$

ForK a field, the ring${\displaystyle K[[X_1,\dots ,X_r]]}$ is often used as the "standard, most general" complete local ring overK in algebra.

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Inmathematical analysis, every convergentpower series defines afunction with values in thereal orcomplex numbers. Formal power series over certain special rings can also be interpreted as functions, but one has to be careful with thedomain andcodomain. Let

${\displaystyle f=\sum a_n{X}^n\in R[[X]],}$

and suppose${\displaystyle S}$ is a commutative associative algebra over${\displaystyle R}$,${\displaystyle I}$ is an ideal in${\displaystyle S}$ such that theI-adic topology on${\displaystyle S}$ is complete, and${\displaystyle x}$ is an element of${\displaystyle I}$. Define:

${\displaystyle f(x)=\sum _{n\ge 0}{a}_n{x}^n.}$

This series is guaranteed to converge in${\displaystyle S}$ given the above assumptions on${\displaystyle x}$. Furthermore, we have

${\displaystyle (f+g)(x)=f(x)+g(x)}$

and

${\displaystyle (fg)(x)=f(x)g(x).}$

Unlike in the case of bona fide functions, these formulas are not definitions but have to be proved.

Since the topology on${\displaystyle R[[X]]}$ is the${\displaystyle (X)}$-adic topology and${\displaystyle R[[X]]}$ is complete, we can in particular apply power series to other power series, provided that the arguments don't haveconstant coefficients (so that they belong to the ideal${\displaystyle (X)}$):${\displaystyle f(0)}$,${\displaystyle f(X^2-X)}$ and${\displaystyle f((1-X{)}^{-1}-1)}$ are all well defined for any formal power series${\displaystyle f\in R[[X]].}$

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

${\displaystyle f^{-1}=\sum _{n\ge 0}{a}^{-n-1}(a-f{)}^n.}$

If the formal power series${\displaystyle g}$ with${\displaystyle g(0)=0}$ is given implicitly by the equation

${\displaystyle f(g)=X}$

where${\displaystyle f}$ is a known power series with${\displaystyle f(0)=0}$, then the coefficients of${\displaystyle g}$ can be explicitly computed using theLagrange inversion formula.

Theformal Laurent series over a ring${\displaystyle R}$ are defined in a similar way to a formal power series, except that we also allow finitely many terms of negative degree. That is, they are the series that can be written as

${\displaystyle f=\sum _{n=N}^{\mathrm{\infty }}{a}_n{X}^n}$

for some integer${\displaystyle N}$, so that there are only finitely many negative${\displaystyle n}$ with${\displaystyle a_n\ne 0}$. (This is different from the classicalLaurent series ofcomplex analysis.) For a non-zero formal Laurent series, the minimal integer${\displaystyle n}$ such that${\displaystyle a_n\ne 0}$ is called theorder of${\displaystyle f}$ and is denoted${\displaystyle \mathrm{ord}(f).}$ (The order ord(0) of the zero series is${\displaystyle +\mathrm{\infty }}$.)

For instance,${\displaystyle X^{-3}+\frac{1}{2}{X}^{-2}+\frac{1}{3}{X}^{-1}+\frac{1}{4}+\frac{1}{5}X+\frac{1}{6}{X}^2+\frac{1}{7}{X}^3+\frac{1}{8}{X}^4+\dots }$ is a formal Laurent series of order –3.

Multiplication of such series can be defined. Indeed, similarly to the definition for formal power series, the coefficient of${\displaystyle X^k}$ of two series with respective sequences of coefficients${\displaystyle \{a_n\}}$ and${\displaystyle \{b_n\}}$ is$${\displaystyle \sum _{i\in \mathbb{Z}}{a}_i{b}_{k-i}.}$$This sum has only finitely many nonzero terms because of the assumed vanishing of coefficients at sufficiently negative indices.

The formal Laurent series form thering of formal Laurent series over${\displaystyle R}$, denoted by${\displaystyle R((X))}$.^[b] It is equal to thelocalization of the ring${\displaystyle R[[X]]}$ of formal power series with respect to the set of positive powers of${\displaystyle X}$. If${\displaystyle R=K}$ is afield, then${\displaystyle K((X))}$ is in fact a field, which may alternatively be obtained as thefield of fractions of theintegral domain${\displaystyle K[[X]]}$.

As with${\displaystyle R[[X]]}$, the ring${\displaystyle R((X))}$ of formal Laurent series may be endowed with the structure of a topological ring by introducing the metric$${\displaystyle d(f,g)=2^{-\mathrm{ord}(f-g)}.}$$(In particular,${\displaystyle \mathrm{ord}(0)=+\mathrm{\infty }}$ implies that${\displaystyle d(f,f)=2^{-\mathrm{ord}(0)}=0}$.)

One may define formal differentiation for formal Laurent series in the natural (term-by-term) way. Precisely, the formal derivative of the formal Laurent series${\displaystyle f}$ above is$${\displaystyle f^{\prime }=Df=\sum _{n\in \mathbb{Z}}n{a}_n{X}^{n-1},}$$which is again a formal Laurent series. If${\displaystyle f}$ is a non-constant formal Laurent series and with coefficients in a field of characteristic 0, then one has$${\displaystyle \mathrm{ord}(f^{\prime })=\mathrm{ord}(f)-1.}$$However, in general this is not the case since the factor${\displaystyle n}$ for the lowest order term could be equal to 0 in${\displaystyle R}$.

Assume that${\displaystyle K}$ is a field ofcharacteristic 0. Then the map

${\displaystyle D:K((X))\to K((X))}$

defined above is a${\displaystyle K}$-derivation that satisfies

${\displaystyle \ker D=K}$

${\displaystyle \mathrm{im}D=\left\{f\in K((X)):[X^{-1}]f=0\right\}.}$

The latter shows that the coefficient of${\displaystyle X^{-1}}$ in${\displaystyle f}$ is of particular interest; it is calledformal residue of${\displaystyle f}$ and denoted${\displaystyle \mathrm{Res}(f)}$. The map

${\displaystyle \mathrm{Res}:K((X))\to K}$

is${\displaystyle K}$-linear, and by the above observation one has anexact sequence

${\displaystyle 0\to K\to K((X))\stackrel{D}{⟶}K((X))\stackrel{\mathrm{Res}}{⟶}K\to 0.}$

Some rules of calculus. As a quite direct consequence of the above definition, and of the rules of formal derivation, one has, for any${\displaystyle f,g\in K((X))}$

1. ${\displaystyle \mathrm{Res}(f^{\prime })=0;}$
2. ${\displaystyle \mathrm{Res}(f{g}^{\prime })=-\mathrm{Res}(f^{\prime }g);}$
3. ${\displaystyle \mathrm{Res}(f^{\prime }/f)=\mathrm{ord}(f),\mathrm{\forall }f\ne 0;}$
4. ${\displaystyle \mathrm{Res}\left((g\circ f)f^{\prime }\right)=\mathrm{ord}(f)\mathrm{Res}(g),}$ if${\displaystyle \mathrm{ord}(f)>0;}$
5. ${\displaystyle [X^n]f(X)=\mathrm{Res}\left(X^{-n-1}f(X)\right).}$

Property (i) is part of the exact sequence above. Property (ii) follows from (i) as applied to${\displaystyle (fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }}$. Property (iii): any${\displaystyle f}$ can be written in the form${\displaystyle f=X^{m}g}$, with${\displaystyle m=\mathrm{ord}(f)}$ and${\displaystyle \mathrm{ord}(g)=0}$: then${\displaystyle f^{\prime }/f=m{X}^{-1}+g^{\prime }/g.}$${\displaystyle \mathrm{ord}(g)=0}$ implies${\displaystyle g}$ is invertible in${\displaystyle K[[X]]\subset \mathrm{im}(D)=\ker (\mathrm{Res}),}$ whence${\displaystyle \mathrm{Res}(f^{\prime }/f)=m.}$ Property (iv): Since${\displaystyle \mathrm{im}(D)=\ker (\mathrm{Res}),}$ we can write${\displaystyle g=g_{-1}{X}^{-1}+G^{\prime },}$ with${\displaystyle G\in K((X))}$. Consequently,${\displaystyle (g\circ f)f^{\prime }=g_{-1}{f}^{-1}{f}^{\prime }+(G^{\prime }\circ f)f^{\prime }=g_{-1}{f}^{\prime }/f+(G\circ f{)}^{\prime }}$ and (iv) follows from (i) and (iii). Property (v) is clear from the definition.

As mentioned above, any formal series${\displaystyle f\in K[[X]]}$ withf₀ = 0 andf₁ ≠ 0 has a composition inverse${\displaystyle g\in K[[X]].}$ The following relation between the coefficients ofgⁿ andf^−k holds ("Lagrange inversion formula"):

${\displaystyle k[X^k]g^n=n[X^{-n}]f^{-k}.}$

In particular, forn = 1 and allk ≥ 1,

${\displaystyle [X^k]g=\frac{1}{k}\mathrm{Res}\left(f^{-k}\right).}$

Since the proof of the Lagrange inversion formula is a very short computation, it is worth reporting one proof here.^[c] Noting${\displaystyle \mathrm{ord}(f)=1}$, we can apply the rules of calculus above, crucially Rule (iv) substituting${\displaystyle X⇝f(X)}$, to get: