Editing Formal power series (section)

===Definition of the formal power series ring===
One can characterize <math>R[[X]]</math> abstractly as the [[completeness (topology)|completion]] of the [[polynomial]] ring <math>R[X]</math> equipped with a particular [[metric space|metric]]<!-- in which the powers of the ideal ''I'' of <math>R[X]</math> generated by <math>X</math> form a shrinking set of neighbourhoods of 0; a precise description would be too long to spell out here-->. This automatically gives <math>R[[X]]</math> the structure of a [[topological 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 <math>R[[X]]</math> more explicitly, and define the ring structure and topological structure separately, as follows.

====Ring structure====
As a set, <math>R[[X]]</math> can be constructed as the set <math>R^\N</math> of all infinite sequences of elements of <math>R</math>, indexed by the [[natural number]]s (taken to include 0). Designating a sequence whose term at index <math>n</math> is <math>a_n</math> by <math>(a_n)</math>, one defines addition of two such sequences by

:<math>(a_n)_{n\in\N} + (b_n)_{n\in\N} = \left( a_n + b_n \right)_{n\in\N}</math>

and multiplication by

:<math>(a_n)_{n\in\N} \times (b_n)_{n\in\N} = \left( \sum_{k=0}^n a_k b_{n-k} \right)_{\!n\in\N}.</math>

This type of product is called the [[Cauchy product]] of the two sequences of coefficients, and is a sort of discrete [[convolution]]. With these operations, <math>R^\N</math> becomes a commutative ring with zero element <math>(0,0,0,\ldots)</math> and multiplicative identity <math>(1,0,0,\ldots)</math>.

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 <math>R</math> into <math>R[[X]]</math> by sending any (constant) <math>a \in R</math> to the sequence <math>(a,0,0,\ldots)</math> and designates the sequence <math>(0,1,0,0,\ldots)</math> by <math>X</math>; then using the above definitions every sequence with only finitely many nonzero terms can be expressed in terms of these special elements as

:<math>(a_0, a_1, a_2, \ldots, a_n, 0, 0, \ldots) = a_0 + a_1 X + \cdots + a_n X^n = \sum_{i=0}^n a_i X^i;</math>

these are precisely the polynomials in <math>X</math>. Given this, it is quite natural and convenient to designate a general sequence <math>(a_n)_{n\in\N}</math> by the formal expression <math>\textstyle\sum_{i\in\N}a_i X^i</math>, 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). This notational convention allows reformulation of the above definitions as

:<math>\left(\sum_{i\in\N} a_i X^i\right)+\left(\sum_{i\in\N} b_i X^i\right) = \sum_{i\in\N}(a_i+b_i) X^i</math>

and

:<math>\left(\sum_{i\in\N} a_i X^i\right) \times \left(\sum_{i\in\N} b_i X^i\right) = \sum_{n\in\N} \left(\sum_{k=0}^n a_k b_{n-k}\right) X^n.</math>

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

==== Topological structure ====
Having stipulated conventionally that

{{NumBlk|:|<math>(a_0, a_1, a_2, a_3, \ldots) = \sum_{i=0}^\infty a_i X^i,</math>|{{EquationRef|1}}}}

one would like to interpret the right hand side as a well-defined infinite summation. To that end, a notion of convergence in <math>R^\N</math> is defined and a [[topology]] on <math>R^\N</math> is constructed. There are several equivalent ways to define the desired topology.

* We may give <math>R^\N</math> the [[product topology]], where each copy of <math>R</math> is given the [[discrete topology]].
* We may give <math>R^\N</math> the [[I-adic topology]], where <math>I=(X)</math> is the ideal generated by <math>X</math>, which consists of all sequences whose first term <math>a_0</math> is zero.
* The desired topology could also be derived from the following [[metric space|metric]]. The distance between distinct sequences <math>(a_n), (b_n) \in R^{\N},</math> is defined to be <math display="block">d((a_n), (b_n)) = 2^{-k},</math> where <math>k</math> is the smallest [[natural number]] such that <math>a_k\neq b_k</math>; the distance between two equal sequences is of course zero.

Informally, two sequences <math>(a_n)</math> and <math>(b_n)</math> become closer and closer if and only if more and more of their terms agree exactly. Formally, the sequence of [[partial sum]]s of some infinite summation converges if for every fixed power of <math>X</math> 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 ({{EquationNote|1}}), regardless of the values <math>a_n</math>, since inclusion of the term for <math>i=n</math> gives the last (and in fact only) change to the coefficient of <math>X^n</math>. It is also obvious that the [[limit of a sequence|limit]] 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 the '''ring of formal power series over <math>R</math>''' and is denoted by <math>R[[X]]</math>. 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 <math>X</math> 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

:<math>\left(\sum_{i\in\N} a_i X^i\right) \times \left(\sum_{i\in\N} b_i X^i\right) = \sum_{i,j\in\N} a_i b_j X^{i+j},</math>

since only finitely many terms on the right affect any fixed <math>X^n</math>. 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).

==== Alternative topologies ====
The above topology is the [[Comparison of topologies|finest topology]] for which

:<math>\sum_{i=0}^\infty a_i X^i</math>

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 <math>R</math> 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 <math>\Z[[X]][[Y]]</math>, the topology of above construction only relates to the indeterminate <math>Y</math>, since the topology that was put on <math>\Z[[X]]</math> has been replaced by the discrete topology when defining the topology of the whole ring. So

:<math>\sum_{i = 0}^\infty XY^i</math>

converges (and its sum can be written as <math>\tfrac{X}{1-Y}</math>); however

:<math>\sum_{i = 0}^\infty X^i Y</math>

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

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 <math>\Z[[X,Y]]</math> and here a sequence converges if and only if the coefficient of every monomial <math>X^iY^j</math> stabilizes. This topology, which is also the <math>I</math>-adic topology, where <math>I=(X,Y)</math> is the ideal generated by <math>X</math> and <math>Y</math>, 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 <math>\R[[X]]</math> the limit

:<math>\lim_{n\to\infty}\left(1+\frac{X}{n}\right)^{\!n}</math>

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

:<math>\exp(X) = \sum_{n\in\N}\frac{X^n}{n!}.</math>

This is because for <math>i\geq 2</math> the coefficient <math>\tbinom{n}{i}/n^i</math> of <math>X^i</math> does not stabilize as <math>n\to \infty</math>. It does however converge in the usual topology of <math>\R</math>, and in fact to the coefficient <math>\tfrac{1}{i!}</math> of <math>\exp(X)</math>. Therefore, if one would give <math>\R[[X]]</math> the product topology of <math>\R^\N</math> where the topology of <math>\R</math> is the usual topology rather than the discrete one, then the above limit would converge to <math>\exp(X)</math>. 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 in [[analysis (mathematics)|analysis]], 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 would ''not'' be the case that a summation converges if and only if its terms tend to 0.