# formal power series

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 $\sum_{n=0}^{\infty}a_{n}X^{n}$, where the coefficients $a_{n}$ are elements of $R$; any choice of coefficients $a_{n}$ is allowed. $R[[X]]$ is actually a topological ring so that these infinite sums are well-defined and convergent   . The addition  and multiplication of such sums follows the usual laws of power series.

## Formal construction

Start with the set $R^{\mathbb{N}}$ of all infinite   sequences in $R$. Define addition of two such sequences by

 $(a_{n})+(b_{n})=(a_{n}+b_{n})$

and multiplication by

 $(a_{n})(b_{n})=(\sum_{k=0}^{n}a_{k}b_{n-k}).$

This turns $R^{\mathbb{N}}$ into a commutative ring with multiplicative identity  (1,0,0,…). We identify the element $a$ of $R$ with the sequence ($a$,0,0,…) and define $X:=(0,1,0,0,\ldots)$. Then every element of $R^{\mathbb{N}}$ of the form $(a_{0},a_{1},a_{2},\ldots,a_{N},0,0,\ldots)$ can be written as the finite sum

 $\sum_{n=0}^{N}a_{n}X^{n}.$

In order to extend this equation to infinite series, we need a metric on $R^{\mathbb{N}}$. We define $d((a_{n}),(b_{n}))=2^{-k}$, where $k$ is the smallest natural number  such that $a_{k}\not=b_{k}$ (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 $R^{\mathbb{N}}$ into a topological ring, and the equation

 $(a_{n})=\sum_{n=0}^{\infty}a_{n}X^{n}$

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

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

## Properties

$R[[X]]$ 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 formula   is valid in $R[[X]]$:

 $(1-X)^{-1}=\sum_{n=0}^{\infty}X^{n}$

An element $\sum a_{n}X^{n}$ of $R[[X]]$ is invertible  in $R[[X]]$ if and only if its constant coefficient $a_{0}$ is invertible in $R$ (see invertible formal power series).  This implies that the Jacobson radical  of $R[[X]]$ is the ideal generated by $X$ and the Jacobson radical of $R$.

Several algebraic  properties of $R$ are inherited by $R[[X]]$:

The metric space $(R[[X]],d)$ is complete     . The topology  on $R[[X]]$ is equal to the product topology on $R^{\mathbb{N}}$ where $R$ is equipped with the discrete topology. It follows from Tychonoff  ’s theorem that $R[[X]]$ is compact  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

 $f=\sum_{n=-M}^{\infty}a_{n}X^{n}$

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

## Formal power series as functions

In analysis  , every convergent power series defines a function with values in the real or complex numbers   . Formal power series can also be interpreted as functions, but one has to be careful with the domain and codomain. If $f=\sum a_{n}X^{n}$ is an element of $R[[X]]$, if $S$ is a commutative   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

 $f(x):=\sum_{n=0}^{\infty}a_{n}x^{n}.$

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

 $(f+g)(x)=f(x)+g(x)$

and

 $(fg)(x)=f(x)g(x)$

(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 arguments   don’t have constant coefficients: $f(0)$, $f(X^{2}-X)$ and $f((1-X)^{-1}-1)$ are all well-defined for any formal power series $f\in R[[X]]$.

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 $R$:

 $f^{-1}=\sum_{n=0}^{\infty}a^{-n-1}(a-f)^{n}$

## Differentiating formal power series

If $f=\sum_{n=0}^{\infty}a_{n}X^{n}\in R[[X]]$, we define the formal derivative of $f$ as

 $\operatorname{D}f=\sum_{n=1}^{\infty}a_{n}nX^{n-1}.$
 $\operatorname{D}(f\cdot g)=(\operatorname{D}f)\cdot g+f\cdot(\operatorname{D}g)$
 $\operatorname{D}(f(g))=(\operatorname{D}f)(g)\cdot\operatorname{D}g$

(in case g(0)=0).

In a sense, all formal power series are Taylor series  , because if $f=\sum a_{n}X^{n}$, then

 $(\operatorname{D}^{k}f)(0)=k!\;a_{k}$

(here $k!$ denotes the element $1\times(1+1)\times(1+1+1)\times\ldots\in R$.

## Power series in several variables

The fastest way to define the ring $R[[X_{1},\ldots,X_{r}]]$ of formal power series over $R$ in $r$ variables starts with the ring $S=R[X_{1},\ldots,X_{r}]$ of polynomials over $R$. Let $I$ be the ideal in $S$ generated by $X_{1},\ldots,X_{r}$, consider the $I$-adic topology on $S$, and form its completion. This results in a complete topological ring containing $S$ which is denoted by $R[[X_{1},\ldots,X_{r}]]$.

For $\mathbf{n}=(n_{1},\ldots,n_{r})\in\mathbb{N}^{r}$, we write $\mathbf{X}^{\mathbf{n}}=X_{1}^{n_{1}}\cdots X_{r}^{n_{r}}$. Then every element of $R[[X_{1},\ldots,X_{r}]]$ can be written in a unique was as a sum

 $\sum_{\mathbf{n}\in\mathbb{N}^{r}}a_{\mathbf{n}}\mathbf{X}^{\mathbf{n}}$

where the sum extends over all $\mathbf{n}\in\mathbb{N}^{r}$. These sums converge for any choice of the coefficients $a_{\mathbf{n}}\in R$ and the order in which the summation is carried out does not matter.

If $J$ is the ideal in $R[[X_{1},\ldots,X_{r}]]$ generated by $X_{1},\ldots,X_{r}$ (i.e. $J$ consists of those power series with zero constant coefficient), then the topology on $R[[X_{1},\ldots,X_{r}]]$ is the $J$-adic topology.

Since $R[[X_{1}]]$ is a commutative ring, we can define its power series ring, say $R[[X_{1}]][[X_{2}]]$. This ring is naturally isomorphic   to the ring $R[[X_{1},X_{2}]]$ just defined, but as topological rings the two are different.

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 derivatives  for formal power series in a straightforward way. Partial derivatives commute, as they do for continuously differentiable functions.

## Uses

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

 $\operatorname{sin}(X):=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}X^{2n+1}$
 $\operatorname{cos}(X):=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}X^{2n}$

Then one can easily show that

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

and

 $D\operatorname{sin}=\operatorname{cos}$

as well as

 $\operatorname{sin}(X+Y)=\operatorname{sin}(X)\operatorname{cos}(Y)+% \operatorname{cos}(X)\operatorname{sin}(Y)$

(the latter being valid in the ring $\mathbb{Q}[[X,Y]]$).

As an example of the method of generating functions, consider the problem of finding a closed formula for the Fibonacci numbers  $f_{n}$ defined by $f_{n+2}=f_{n+1}+f_{n}$, $f_{0}=0$, and $f_{1}=1$. We work in the ring $\mathbb{R}[[X]]$ and define the power series

 $f=\sum_{n=0}^{\infty}f_{n}X^{n};$

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

 $f=Xf+X^{2}f+X$

(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

 $f=\frac{X}{1-X-X^{2}}.$

Using the golden ratio  $\phi_{1}=(1+\sqrt{5})/2$ and $\phi_{2}=(1-\sqrt{5})/2$, we can write the latter expression as

 $\frac{1}{\sqrt{5}}\left(\frac{1}{1-\phi_{1}X}-\frac{1}{1-\phi_{2}X}\right).$

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

 $f_{n}=\frac{1}{\sqrt{5}}\left(\phi_{1}^{n}-\phi_{2}^{n}\right).$

In algebra, the ring $K[[X_{1},\ldots,X_{r}]]$ (where $K$ is a field) is often used as the “standard, most general” complete local ring over $K$.

## Universal property

The power series ring $R[[X_{1},\ldots,X_{r}]]$ 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 $x_{1},\ldots,x_{r}\in I$ are given, then there exists a unique $\Phi:R[[X_{1},\ldots,X_{r}]]\to S$ with the following properties:

 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