Editing Formal power series (section)

==== 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).