{"history":[],"article":{"id":"2283a8fe-eac1-4f31-b4ae-56ce79315b5e","title":"Formal power series","slug":"formal-power-series","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a **formal [series](https://bliptext.com/articles/series-mathematics)** is an infinite sum that is considered independently from any notion of [convergence](https://bliptext.com/articles/convergent-[series](https://bliptext.com/articles/series-mathematics)), and can be manipulated with the usual algebraic operations on [series](https://bliptext.com/articles/series-mathematics) (addition, subtraction, multiplication, division, [partial sums](https://bliptext.com/articles/partial-sum), etc.).\nA **formal power series** is a special kind of formal series, of the form\nwhere the a_n, called coefficients, are numbers or, more generally, elements of some [ring](https://bliptext.com/articles/ring-mathematics), and the x^n are formal powers of the symbol x that is called an [indeterminate](https://bliptext.com/articles/indeterminate-[variable](https://bliptext.com/articles/variable-mathematics)) or, commonly, a [variable](https://bliptext.com/articles/variable-mathematics).\nHence, [power series](https://bliptext.com/articles/power-series) can be viewed as a generalization of [polynomials](https://bliptext.com/articles/polynomial) where the number of terms is allowed to be infinite, and differ from usual [power series](https://bliptext.com/articles/power-series) by the absence of convergence requirements, which implies that a [power series](https://bliptext.com/articles/power-series) may not represent a function of its variable.\nFormal power series are in [one to one correspondence](https://bliptext.com/articles/one-to-one-correspondence) with their [sequences](https://bliptext.com/articles/sequence-mathematics) of coefficients, but the two concepts must not be confused, since the operations that can be applied are different.\nA formal power series with coefficients in a ring R is called a formal power series over R. The formal power series over a ring R form a ring, commonly denoted by Rx.\n(It can be seen as the [[I-adic topology#Completion|\n(x)\n-adic completion]] of the [polynomial ring](https://bliptext.com/articles/polynomial-ring) R[x], in the same way as the [p-adic integers](https://bliptext.com/articles/p-adic-integer) are the p-adic completion of the ring of the integers.)\nFormal powers series in several indeterminates are defined similarly by replacing the powers of a single indeterminate by [monomials](https://bliptext.com/articles/monomial) in several indeterminates.\nFormal power series are widely used in [combinatorics](https://bliptext.com/articles/combinatorics) for representing sequences of integers as [generating functions](https://bliptext.com/articles/generating-function).\nIn this context, a [recurrence relation](https://bliptext.com/articles/recurrence-relation) between the elements of a sequence may often be interpreted as a [differential equation](https://bliptext.com/articles/differential-equation) that the generating function satisfies.\nThis allows using methods of [complex analysis](https://bliptext.com/articles/complex-analysis) for combinatorial problems (see [analytic combinatorics](https://bliptext.com/articles/analytic-combinatorics)).\n\n# Introduction\nA formal power series can be loosely thought of as an object that is like a [polynomial](https://bliptext.com/articles/polynomial), but with infinitely many terms.\nAlternatively, for those familiar with [power series](https://bliptext.com/articles/power-series) (or [Taylor series](https://bliptext.com/articles/taylor-series)), one may think of a formal [power series](https://bliptext.com/articles/power-series) as a [power series](https://bliptext.com/articles/power-series) in which we ignore questions of [convergence](https://bliptext.com/articles/convergent-series) by not assuming that the variable X denotes any numerical value (not even an unknown value).\nFor example, consider the series\nIf we studied this as a power series, its properties would include, for example, that its [radius of convergence](https://bliptext.com/articles/radius-of-convergence) is 1 by the [Cauchy–Hadamard theorem](https://bliptext.com/articles/cauchy-hadamard-theorem).\nHowever, as a formal power series, we may ignore this completely; all that is relevant is the sequence of [coefficients](https://bliptext.com/articles/coefficient) [1, −3, 5, −7, 9, −11, ...].\nIn other words, a formal power series is an object that just records a sequence of coefficients.\nIt is perfectly acceptable to consider a formal power series with the [factorials](https://bliptext.com/articles/factorial) [1, 1, 2, 6, 24, 120, 720, 5040, ... ] as coefficients, even though the corresponding power series diverges for any nonzero value of X.\nAlgebra on formal power series is carried out by simply pretending that the series are polynomials.\nFor example, if\nthen we add A and B term by term:\nWe can multiply formal power series, again just by treating them as polynomials (see in particular [Cauchy product](https://bliptext.com/articles/cauchy-product)):\nNotice that each coefficient in the product AB only depends on a finite number of coefficients of A and B.\nFor example, the X5 term is given by\nFor this reason, one may multiply formal power series without worrying about the usual questions of [absolute](https://bliptext.com/articles/absolute-convergence), [conditional](https://bliptext.com/articles/conditional-convergence) and [uniform convergence](https://bliptext.com/articles/uniform-convergence) which arise in dealing with power series in the setting of [analysis](https://bliptext.com/articles/mathematical-analysis).\nOnce we have defined multiplication for formal power series, we can define multiplicative inverses as follows.\nThe 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.\nIt turns out that if A has a multiplicative inverse, it is unique, and we denote it by A−1.\nNow 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.\nFor example, one can use the definition of multiplication above to verify the familiar formula\nAn important operation on formal power series is coefficient extraction.\nIn its most basic form, the coefficient extraction operator [X^n] applied to a formal power series A in one variable extracts the coefficient of the nth power of the variable, so that [X^2]A=5 and [X^5]A=-11.\nOther examples include\nSimilarly, many other operations that are carried out on polynomials can be extended to the formal power series setting, as explained below.\n\n# The ring of formal power series\nIf one considers the set of all formal power series in X with coefficients in a [commutative ring](https://bliptext.com/articles/commutative-ring) R, the elements of this set collectively constitute another ring which is written RX, and called the **ring of formal power series** in the variable X over R.\n\n## Definition of the formal power series ring\nOne can characterize RX abstractly as the [completion](https://bliptext.com/articles/completeness-topology) of the [polynomial](https://bliptext.com/articles/polynomial) ring R[X] equipped with a particular [metric](https://bliptext.com/articles/metric-space).\nThis automatically gives RX the structure of a [topological ring](https://bliptext.com/articles/topological-ring) (and even of a complete metric space).\nBut 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.\nIt is possible to describe RX more explicitly, and define the ring structure and topological structure separately, as follows.\n\n### Ring structure\nAs a set, RX can be constructed as the set R^\\N of all infinite sequences of elements of R, indexed by the [natural numbers](https://bliptext.com/articles/natural-number) (taken to include 0).\nDesignating a sequence whose term at index n is a_n by (a_n), one defines addition of two such sequences by\nand multiplication by\nThis type of product is called the [Cauchy product](https://bliptext.com/articles/cauchy-product) of the two sequences of coefficients, and is a sort of discrete [convolution](https://bliptext.com/articles/convolution).\nWith these operations, R^\\N becomes a commutative ring with zero element and multiplicative identity.\nThe product is in fact the same one used to define the product of polynomials in one indeterminate, which suggests using a similar notation.\nOne embeds R into RX by sending any (constant) a \\in R to the sequence and designates the sequence by X; then using the above definitions every sequence with only finitely many nonzero terms can be expressed in terms of these special elements as\nthese are precisely the polynomials in X. Given this, it is quite natural and convenient to designate a general sequence by the formal expression, 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).\nThis notational convention allows reformulation of the above definitions as\nand\nwhich is quite convenient, but one must be aware of the distinction between formal summation (a mere convention) and actual addition.\n\n### Topological structure\nHaving stipulated conventionally that\none would like to interpret the right hand side as a well-defined infinite summation.\nTo that end, a notion of convergence in R^\\N is defined and a [topology](https://bliptext.com/articles/topology) on R^\\N is constructed.\nThere are several equivalent ways to define the desired topology.\nInformally, two sequences (a_n) and (b_n) become closer and closer if and only if more and more of their terms agree exactly.\nFormally, the sequence of [partial sums](https://bliptext.com/articles/partial-sum) of some infinite summation converges if for every fixed power of X the coefficient stabilizes: there is a point beyond which all further [partial sums](https://bliptext.com/articles/partial-sum) have the same coefficient.\nThis is clearly the case for the right hand side of, regardless of the values a_n, since inclusion of the term for i=n gives the last (and in fact only) change to the coefficient of X^n.\nIt is also obvious that the [limit](https://bliptext.com/articles/limit-of-a-sequence) of the sequence of partial sums is equal to the left hand side.\nThis topological structure, together with the ring operations described above, form a topological ring.\nThis is called the **ring of formal power series over R** and is denoted by RX.\nThe 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 X occurs in only finitely many terms.\nThe topological structure allows much more flexible usage of infinite summations.\nFor instance the rule for multiplication can be restated simply as\nsince only finitely many terms on the right affect any fixed X^n.\nInfinite 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).\n\n### Alternative topologies\nThe above topology is the [finest topology](https://bliptext.com/articles/comparison-of-topologies) for which\nalways 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.\nIt can however happen occasionally that one wishes to use a coarser topology, so that certain expressions become convergent that would otherwise diverge.\nThis applies in particular when the base ring R already comes with a topology other than the discrete one, for instance if it is also a ring of formal power series.\nIn the ring of formal power series \\ZXY, the topology of above construction only relates to the indeterminate Y, since the topology that was put on \\ZX has been replaced by the discrete topology when defining the topology of the whole ring.\nSo\nconverges (and its sum can be written as ); however\nwould be considered to be divergent, since every term affects the coefficient of Y. This asymmetry disappears if the power series ring in Y is given the product topology where each copy of \\ZX is given its topology as a ring of formal power series rather than the discrete topology.\nWith this topology, a sequence of elements of \\ZXY converges if the coefficient of each power of Y converges to a formal power series in X, a weaker condition than stabilizing entirely.\nFor instance, with this topology, in the second example given above, the coefficient of Yconverges to, so the whole summation converges to.\nThis 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.\nIn the above example that would mean constructing \\ZX,Y and here a sequence converges if and only if the coefficient of every monomial X^iY^j stabilizes.\nThis topology, which is also the I-adic topology, where I=(X,Y) is the ideal generated by X and Y, still enjoys the property that a summation converges if and only if its terms tend to 0.\nThe same principle could be used to make other divergent limits converge.\nFor instance in \\RX the limit\ndoes not exist, so in particular it does not converge to\nThis is because for i\\geq 2 the coefficient of X^i does not stabilize as n\\to \\infty.\nIt does however converge in the usual topology of \\R, and in fact to the coefficient of \\exp(X).\nTherefore, if one would give \\RX the product topology of \\R^\\N where the topology of \\R is the usual topology rather than the discrete one, then the above limit would converge to \\exp(X).\nThis 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](https://bliptext.com/articles/analysis-mathematics), while the philosophy of formal power series is on the contrary to make convergence questions as trivial as they can possibly be.\nWith this topology it would not be the case that a summation converges if and only if its terms tend to 0.\n\n## Universal property\nThe ring RX may be characterized by the following [universal property](https://bliptext.com/articles/universal-property).\nIf 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 x is an element of I, then there is a unique with the following properties:\n\n# Operations on formal power series\nOne can perform algebraic operations on power series to generate new power series.\nBesides the ring structure operations defined above, we have the following.\n\n## Power series raised to powers\nFor any [natural number](https://bliptext.com/articles/natural-number)\nn\n, the nth power of a formal power series S is defined recursively by\nIf\nm\nand\na0\nare invertible in the ring of coefficients, one can prove\nwhere\nIn the case of formal power series with complex coefficients, the complex powers are well defined for series\nf\nwith constant term equal to\n1\n. In this case, f^{\\alpha} can be defined either by composition with the [binomial series](https://bliptext.com/articles/binomial-series)\n(1+x)α\n, or by composition with the exponential and the logarithmic series, or as the solution of the differential equation (in terms of series) with constant term 1; the three definitions are equivalent.\nThe rules of calculus and easily follow.\n\n## Multiplicative inverse\nThe series\nis invertible in RX if and only if its constant coefficient a_0 is invertible in R. This condition is necessary, for the following reason: if we suppose that A has an inverse then the [constant term](https://bliptext.com/articles/constant-term) a_0b_0 of A \\cdot B is the [constant term](https://bliptext.com/articles/constant-term) of the identity series, i.e. it is 1.\nThis condition is also sufficient; we may compute the coefficients of the inverse series B via the explicit recursive formula\nAn important special case is that the [geometric series](https://bliptext.com/articles/geometric-series) formula is valid in RX:\nIf 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 X. This means that KX is a [discrete valuation ring](https://bliptext.com/articles/discrete-valuation-ring) with uniformizing parameter X.\n\n## Division\nThe computation of a quotient f/g=h\nassuming the denominator is invertible (that is, a_0 is invertible in the ring of scalars), can be performed as a product f and the inverse of g, or directly equating the coefficients in f=gh:\n\n## Extracting coefficients\nThe coefficient extraction operator applied to a formal power series\nin X is written\nand extracts the coefficient of Xm, so that\n\n## Composition\nGiven two formal power series\nsuch that a_0=0,\none may form the composition\nwhere the coefficients cn are determined by \"expanding out\" the powers of f(X):\nHere the sum is extended over all (k, j) with k\\in\\N and j\\in\\N_+^k with\nSince a_0=0, one must have k\\le n and j_i\\le n for every i. This implies that the above sum is finite and that the coefficient c_n is the coefficient of X^n in the polynomial g_n(f_n(X)), where f_n and g_n are the polynomials obtained by truncating the series at x^n, that is, by removing all terms involving a power of X higher than n.\nA more explicit description of these coefficients is provided by [Faà di Bruno's formula](https://bliptext.com/articles/fa-di-bruno-s-formula), at least in the case where the coefficient ring is a field of [characteristic 0](https://bliptext.com/articles/characteristic-algebra).\nComposition is only valid when f(X) has no constant term, so that each c_n depends on only a finite number of coefficients of f(X) and g(X).\nIn other words, the series for g(f(X)) converges in the [topology](https://bliptext.com/articles/completion-ring-theory) of RX.\n\n### Example\nAssume that the ring R has characteristic 0 and the nonzero integers are invertible in R. If one denotes by \\exp(X) the formal power series\nthen the equality\nmakes perfect sense as a formal power series, since the constant coefficient of \\exp(X) - 1 is zero.\n\n## Composition inverse\nWhenever a formal series\nhas f0 = 0 and f1 being an invertible element of R, there exists a series\nthat is the [composition inverse](https://bliptext.com/articles/composition-inverse) of f, meaning that composing f with g gives the series representing the [identity function](https://bliptext.com/articles/identity-function).\nThe coefficients of g may be found recursively by using the above formula for the coefficients of a composition, equating them with those of the composition identity X (that is 1 at degree 1 and 0 at every degree greater than 1).\nIn the case when the coefficient ring is a field of characteristic 0, the [Lagrange inversion formula](https://bliptext.com/articles/) (discussed below) provides a powerful tool to compute the coefficients of g, as well as the coefficients of the (multiplicative) powers of g.\n\n## Formal differentiation\nGiven a formal power series\nwe define its **[formal derivative](https://bliptext.com/articles/formal-derivative)**, denoted Df or f ′, by\nThe symbol D is called the **formal differentiation operator**.\nThis definition simply mimics term-by-term differentiation of a polynomial.\nThis operation is R-[linear](https://bliptext.com/articles/linear-operator):\nfor any a, b in R and any f, g in RX.\nAdditionally, the formal [derivative](https://bliptext.com/articles/derivative) has many of the properties of the usual [derivative](https://bliptext.com/articles/derivative) of calculus.\nFor example, the [product rule](https://bliptext.com/articles/product-rule) is valid:\nand the [chain rule](https://bliptext.com/articles/chain-rule) works as well:\nwhenever the appropriate compositions of series are defined (see above under [composition of series](https://bliptext.com/articles/)).\nThus, in these respects formal power series behave like [Taylor series](https://bliptext.com/articles/taylor-series).\nIndeed, for the f defined above, we find that\nwhere Dk denotes the kth formal derivative (that is, the result of formally differentiating k times).\n\n## Formal antidifferentiation\nIf R is a ring with characteristic zero and the nonzero integers are invertible in R, then given a formal power series\nwe define its **formal antiderivative** or **formal indefinite integral** by\nfor any constant C \\in R.\nThis operation is R-[linear](https://bliptext.com/articles/linear-operator):\nfor any a, b in R and any f, g in RX.\nAdditionally, the formal [antiderivative](https://bliptext.com/articles/antiderivative) has many of the properties of the usual [antiderivative](https://bliptext.com/articles/antiderivative) of calculus.\nFor example, the formal antiderivative is the [right inverse](https://bliptext.com/articles/inverse-function) of the formal derivative:\nfor any f \\in RX.\n\n# Properties\n\n## Algebraic properties of the formal power series ring\nRX is an [associative algebra](https://bliptext.com/articles/associative-algebra) over R which contains the ring R[X] of polynomials over R; the polynomials correspond to the sequences which end in zeros.\nThe [Jacobson radical](https://bliptext.com/articles/jacobson-radical) of RX is the [ideal](https://bliptext.com/articles/ideal-ring-theory) generated by X and the [Jacobson radical](https://bliptext.com/articles/jacobson-radical) of R; this is implied by the element invertibility criterion discussed above.\nThe [maximal ideals](https://bliptext.com/articles/maximal-ideal) of RX all arise from those in R in the following manner: an ideal M of RX is maximal if and only if M\\cap R is a maximal ideal of R and M is generated as an ideal by X and M\\cap R.\nSeveral algebraic properties of R are inherited by RX:\n\n## Topological properties of the formal power series ring\nThe metric space (RX, d) is [complete](https://bliptext.com/articles/completeness-topology).\nThe ring RX is [compact](https://bliptext.com/articles/compact-space) if and only if R is [finite](https://bliptext.com/articles/finite-set).\nThis follows from [Tychonoff's theorem](https://bliptext.com/articles/tychonoff-s-theorem) and the characterisation of the topology on RX as a product topology.\n\n## Weierstrass preparation\nThe ring of formal power series with coefficients in a [complete local ring](https://bliptext.com/articles/complete-local-ring) satisfies the [Weierstrass preparation theorem](https://bliptext.com/articles/weierstrass-preparation-theorem).\n\n# Applications\nFormal power series can be used to solve recurrences occurring in number theory and combinatorics.\nFor an example involving finding a closed form expression for the [Fibonacci numbers](https://bliptext.com/articles/fibonacci-number), see the article on [Examples of generating functions](https://bliptext.com/articles/examples-of-generating-functions).\nOne can use formal power series to prove several relations familiar from analysis in a purely algebraic setting.\nConsider for instance the following elements of \\QX:\nThen one can show that\nThe last one being valid in the ring \\QX, Y.\nFor K a field, the ring is often used as the \"standard, most general\" complete local ring over K in algebra.\n\n# Interpreting formal power series as functions\nIn [mathematical analysis](https://bliptext.com/articles/mathematical-analysis), every convergent [power series](https://bliptext.com/articles/power-series) defines a [function](https://bliptext.com/articles/function-mathematics) with values in the [real](https://bliptext.com/articles/real-number) or [complex](https://bliptext.com/articles/complex-number) numbers.\nFormal power series over certain special rings can also be interpreted as functions, but one has to be careful with the [domain](https://bliptext.com/articles/function-domain) and co[domain](https://bliptext.com/articles/function-domain).\nLet\nand suppose S is a commutative associative algebra over R, I is an ideal in S such that the [I-adic topology](https://bliptext.com/articles/i-adic-topology) on S is complete, and x is an element of I. Define:\nThis series is guaranteed to converge in S given the above assumptions on x. Furthermore, we have\nand\nUnlike in the case of bona fide functions, these formulas are not definitions but have to be proved.\nSince the topology on RX is the (X)-adic topology and RX is complete, we can in particular apply power series to other power series, provided that the arguments don't have [constant coefficients](https://bliptext.com/articles/constant-coefficients) (so that they belong to the ideal (X)): f(0), f(X^2-X) and are all well defined for any formal power series f \\in RX.\nWith 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:\nIf the formal power series g with g(0)=0 is given implicitly by the equation\nwhere f is a known power series with f(0)=0, then the coefficients of g can be explicitly computed using the [Lagrange inversion formula](https://bliptext.com/articles/).\n\n# Generalizations\n\n## Formal Laurent series\nThe **formal Laurent series** over a ring R are defined in a similar way to a formal power series, except that we also allow finitely many terms of negative degree.\nThat is, they are the series that can be written as\nfor some integer N, so that there are only finitely many negative n with a_n \\neq 0.\n(This is different from the classical [Laurent series](https://bliptext.com/articles/laurent-series) of [complex analysis](https://bliptext.com/articles/complex-analysis).) For a non-zero formal [Laurent series](https://bliptext.com/articles/laurent-series), the minimal integer n such that a_n\\neq 0 is called the order of f and is denoted (The order of the zero series is +\\infty.)\nMultiplication of such series can be defined.\nIndeed, similarly to the definition for formal power series, the coefficient of X^k of two series with respective sequences of coefficients \\{a_n\\} and \\{b_n\\} is\nThis sum has only finitely many nonzero terms because of the assumed vanishing of coefficients at sufficiently negative indices.\nThe formal Laurent series form the **ring of formal Laurent series** over R, denoted by R((X)).\nIt is equal to the [localization](https://bliptext.com/articles/localization-of-a-ring) of the ring RX of formal power series with respect to the set of positive powers of X. If R=K is a [field](https://bliptext.com/articles/field-mathematics), then K((X)) is in fact a [field](https://bliptext.com/articles/field-mathematics), which may alternatively be obtained as the [field](https://bliptext.com/articles/field-mathematics) of fractions of the [integral domain](https://bliptext.com/articles/integral-domain) KX.\nAs with RX, the ring R((X)) of formal Laurent series may be endowed with the structure of a topological ring by introducing the metric\nOne may define formal differentiation for formal Laurent series in the natural (term-by-term) way.\nPrecisely, the formal derivative of the formal Laurent series f above is\nwhich is again a formal Laurent series.\nIf f is a non-constant formal Laurent series and with coefficients in a field of characteristic 0, then one has\nHowever, in general this is not the case since the factor n for the lowest order term could be equal to 0 in R.\n\n### Formal residue\nAssume that K is a field of [characteristic](https://bliptext.com/articles/characteristic-algebra) 0.\nThen the map\ndefined above is a K-[derivation](https://bliptext.com/articles/derivation-abstract-algebra) that satisfies\nThe latter shows that the coefficient of X^{-1} in f is of particular interest; it is called formal residue of f and denoted.\nThe map\nis K-linear, and by the above observation one has an [exact sequence](https://bliptext.com/articles/exact-sequence)\n**Some rules of calculus**.\nAs a quite direct consequence of the above definition, and of the rules of formal derivation, one has, for any\n

\nProperty (i) is part of the exact sequence above.\nProperty (ii) follows from (i) as applied to.\nProperty (iii): any f can be written in the form f=X^mg, with and : then implies g is invertible in whence Property (iv): Since we can write with.\nConsequently, and (iv) follows from (i) and (iii).\nProperty (v) is clear from the definition.\n\n## The Lagrange inversion formula\nAs mentioned above, any formal series f \\in KX with f0 = 0 and f1 ≠ 0 has a composition inverse g \\in KX.\nThe following relation between the coefficients of gn and f−k holds (\"\"):\nIn particular, for n = 1 and all k ≥ 1,\nSince the proof of the Lagrange inversion formula is a very short computation, it is worth reporting one residue-based proof here (a number of different proofs exist, using, e.g., Cauchy's coefficient formula for holomorphic functions, tree-counting arguments, or induction).\nNoting, we can apply the rules of calculus above, crucially Rule (iv) substituting , to get:\n**Generalizations.** One may observe that the above computation can be repeated plainly in more general settings than K((X)): a generalization of the Lagrange inversion formula is already available working in the -modules where α is a complex exponent.\nAs a consequence, if f and g are as above, with f_1=g_1=1, we can relate the complex powers of f / X and g / X: precisely, if α and β are non-zero complex numbers with negative integer sum, then\nFor instance, this way one finds the power series for [complex powers of the Lambert function](https://bliptext.com/articles/lambert-w-function).\n\n## Power series in several variables\nFormal power series in any number of indeterminates (even infinitely many) can be defined.\nIf I is an index set and XI is the set of indeterminates Xi for i∈I, then a [monomial](https://bliptext.com/articles/monomial) Xα is any finite product of elements of XI (repetitions allowed); a formal power series in XI with coefficients in a ring R is determined by any mapping from the set of [monomial](https://bliptext.com/articles/monomial)s Xα to a corresponding coefficient cα, and is denoted.\nThe set of all such formal power series is denoted RX_I, and it is given a ring structure by defining\nand\n\n### Topology\nThe topology on RX_I is such that a sequence of its elements converges only if for each monomial Xα the corresponding coefficient stabilizes.\nIf I is finite, then this the J-adic topology, where J is the ideal of RX_I generated by all the indeterminates in XI.\nThis does not hold if I is infinite.\nFor example, if I=\\N, then the sequence with does not converge with respect to any J-adic topology on R, but clearly for each monomial the corresponding coefficient stabilizes.\nAs remarked above, the topology on a repeated formal power series ring like RXY is usually chosen in such a way that it becomes isomorphic as a [topological ring](https://bliptext.com/articles/topological-ring) to RX,Y.\n\n### Operations\nAll of the operations defined for series in one variable may be extended to the several variables case.\nIn the case of the formal derivative, there are now separate [partial derivative](https://bliptext.com/articles/partial-derivative) operators, which differentiate with respect to each of the indeterminates.\nThey all commute with each other.\n\n### Universal property\nIn the several variables case, the universal property characterizing becomes the following.\nIf 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 with the following properties:\n\n## Non-commuting variables\nThe several variable case can be further generalised by taking non-commuting variables Xi for i ∈ I, where I is an index set and then a [monomial](https://bliptext.com/articles/monomial) Xα is any [word](https://bliptext.com/articles/word-mathematics) in the XI; a formal power series in XI with coefficients in a ring R is determined by any mapping from the set of [monomial](https://bliptext.com/articles/monomial)s Xα to a corresponding coefficient cα, and is denoted.\nThe set of all such formal power series is denoted R«XI», and it is given a ring structure by defining addition pointwise\nand multiplication by\nwhere · denotes concatenation of words.\nThese formal power series over R form the **Magnus ring** over R.\n\n## On a semiring\nGiven an [alphabet](https://bliptext.com/articles/alphabet-formal-languages) \\Sigma and a [semiring](https://bliptext.com/articles/semiring) S. The formal power series over S supported on the language \\Sigma^* is denoted by.\nIt consists of all mappings, where \\Sigma^* is the [free monoid](https://bliptext.com/articles/free-monoid) generated by the non-empty set \\Sigma.\nThe elements of can be written as formal sums\nwhere (r,w) denotes the value of r at the word.\nThe elements (r,w)\\in S are called the coefficients of r.\nFor the support of r is the set\nA series where every coefficient is either 0 or 1 is called the characteristic series of its support.\nThe subset of consisting of all series with a finite support is denoted by and called polynomials.\nFor and s\\in S, the sum r_1+r_2 is defined by The (Cauchy) product is defined by The Hadamard product is defined by And the products by a scalar sr_1 and r_1s by\nWith these operations and are semirings, where \\varepsilon is the empty word in \\Sigma^*.\nThese formal power series are used to model the behavior of [weighted automata](https://bliptext.com/articles/weighted-automata), in [theoretical computer science](https://bliptext.com/articles/theoretical-computer-science), when the coefficients (r,w) of the series are taken to be the weight of a path with label w in the automata.\n\n## Replacing the index set by an ordered abelian group\nSuppose G is an ordered abelian group, meaning an abelian group with a total ordering < respecting the group's addition, so that a

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