Editing Formal power series (section)

=== Power series in several variables ===
Formal power series in any number of indeterminates (even infinitely many) can be defined. If ''I'' is an index set and ''X<sub>I</sub>'' is the set of indeterminates ''X<sub>i</sub>'' for ''i''∈''I'', then a [[monomial]] ''X''<sup>''α''</sup> is any finite product of elements of ''X<sub>I</sub>'' (repetitions allowed); a formal power series in ''X<sub>I</sub>'' with coefficients in a ring ''R'' is determined by any mapping from the set of monomials ''X''<sup>''α''</sup> to a corresponding coefficient ''c''<sub>''α''</sub>, and is denoted <math display="inline">\sum_\alpha c_\alpha X^\alpha</math>. The set of all such formal power series is denoted <math>R[[X_I]],</math> and it is given a ring structure by defining

:<math>\left(\sum_\alpha c_\alpha X^\alpha\right)+\left(\sum_\alpha d_\alpha X^\alpha \right)= \sum_\alpha (c_\alpha+d_\alpha) X^\alpha</math>

and

:<math>\left(\sum_\alpha c_\alpha X^\alpha\right)\times\left(\sum_\beta d_\beta X^\beta\right)=\sum_{\alpha,\beta} c_\alpha d_\beta X^{\alpha+\beta}</math>

==== Topology ====
The topology on <math>R[[X_I]]</math> is such that a sequence of its elements converges only if for each monomial ''X''<sup>α</sup> the corresponding coefficient stabilizes. If ''I'' is finite, then this the ''J''-adic topology, where ''J'' is the ideal of <math>R[[X_I]]</math> generated by all the indeterminates in ''X<sub>I</sub>''. This does not hold if ''I'' is infinite. For example, if <math>I=\N,</math> then the sequence <math>(f_n)_{n\in \N}</math> with <math>f_n = X_n + X_{n+1} + X_{n+2} + \cdots </math> does not converge with respect to any ''J''-adic topology on ''R'', but clearly for each monomial the corresponding coefficient stabilizes.

As remarked above, the topology on a repeated formal power series ring like <math>R[[X]][[Y]]</math> is usually chosen in such a way that it becomes isomorphic as a [[topological ring]] to <math>R[[X,Y]].</math>

====Operations====
All of the operations defined for series in one variable may be extended to the several variables case.

* A series is invertible if and only if its constant term is invertible in ''R''.
* The composition ''f''(''g''(''X'')) of two series ''f'' and ''g'' is defined if ''f'' is a series in a single indeterminate, and the constant term of ''g'' is zero. For a series ''f'' in several indeterminates a form of "composition" can similarly be defined, with as many separate series in the place of ''g'' as there are indeterminates.

In the case of the formal derivative, there are now separate [[partial derivative]] operators, which differentiate with respect to each of the indeterminates. They all commute with each other.

==== Universal property ====
In the several variables case, the universal property characterizing <math>R[[X_1, \ldots, X_r]]</math> becomes the following. If ''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''<sub>1</sub>, ..., ''x<sub>r</sub>'' are elements of ''I'', then there is a ''unique'' map <math>\Phi: R[[X_1, \ldots, X_r]] \to S</math> with the following properties:

* Φ is an ''R''-algebra homomorphism
* Φ is continuous
* Φ(''X''<sub>''i''</sub>) = ''x''<sub>''i''</sub> for ''i'' = 1, ..., ''r''.