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Formal power series allow one to employ much of the analytical machinery of power series in settings which don't have natural notions of convergence. They are also useful in order to compactly describe sequences 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 $\sum_{n=0}^\infty a_n X^n$ , 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.
Start with the set $R^\Bbb{N}$ of all infinite sequences in $R$ . Define addition of two such sequences by $$(a_n) + (b_n) = (a_n + b_n)$$ and multiplication by $$(a_n) (b_n) = (\sum_{k=0}^n a_kb_{n-k}).$$ This turns $R^\Bbb{N}$ 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 $R^\Bbb{N}$ of the form $(a_0,a_1,a_2,\ldots,a_N,0,0,\ldots)$ can be written as the finite sum $$\sum_{n=0}^Na_n X^n.$$ In order to extend this equation to infinite series, we need a metric on $R^\Bbb{N}$ . 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 $R^\Bbb{N}$ 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 $R[[X]]$ .
$R[[X]]$ 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 $R^\Bbb{N}$ 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
$f$ .
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 $R$ : $$f^{-1} = \sum_{n=0}^\infty a^{-n-1} (a-f)^n$$
If $f=\sum_{n=0}^\infty a_n X^n\in R[[X]]$ , we define the formal derivative of $f$ as $$\operatorname{D}f = \sum_{n=1}^{\infty} a_n n X^{n-1}.$$ This operation is $R$ -linear, obeys the product rule $$\operatorname{D}(f\cdot g) = (\operatorname{D}f)\cdot g + f\cdot (\operatorname{D} g)$$ and the chain rule: $$\operatorname{D}(f(g)) = (\operatorname{D}f)(g)\cdot \operatorname{D}g$$ (in case g(0)=0).
In a sense, all formal power series are Taylor series, because if $f=\sum a_n X^n$ , then $$(\operatorname{D}^kf)(0) = k!\; a_k$$ (here $k!$ denotes the element $1\times (1+1)\times(1+1+1)\times\ldots\in R$ .
One can also define differentiation for formal Laurent series in a natural way, and then the quotient rule, in addition to the rules listed above, will also be valid.
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 $\mathbf{n}=(n_1,\ldots,n_r)\in\Bbb{N}^r$ , 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 $$\sum_{\mathbf{n}\in\Bbb{N}^r} a_{\mathbf{n}} \mathbf{X}^\mathbf{n}$$ where the sum extends over all $\mathbf{n}\in\Bbb{N}^r$ . 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.
If $K=R$ is a field, then $K[[X_1,\ldots,X_r]]$ is a unique factorization domain.
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.
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 $\Bbb{Q}[[X]]$ : $$\operatorname{sin}(X) := \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} X^{2n+1}$$ $$\operatorname{cos}(X) := \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!} X^{2n}$$ 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 $\Bbb{Q}[[X,Y]]$ ).
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 $\Bbb{R}[[X]]$ 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 $X^2f$ . From the recurrence relation, we therefore see that the power series $Xf + X^2f$ agrees with $f$ except for the first two coefficients. Taking these into account, we find that $$f=Xf+X^2f+X$$ (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 $$\frac{1}{\sqrt{5}}\left(\frac{1}{1-\phi_1X}-\frac{1}{1-\phi_2X}\right).$$ 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 $K$ .
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:
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