Editing Formal power series (section)

====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.