# 形式幂级数

## 简介

${\displaystyle A=1-3X+5X^{2}-7X^{3}+9X^{4}-11X^{5}+\cdots .}$

${\displaystyle A=1+X+2X^{2}+6X^{3}+24X^{4}+120X^{5}+\cdots .}$

${\displaystyle X}$取任何的非零实数值时都不收敛，我们仍然可以将其作为形式幂级数进行运算。

${\displaystyle B=2X+4X^{3}+6X^{5}+8X^{7}+\cdots .}$

${\displaystyle A+B=1+3X+2X^{2}+10X^{3}+24X^{4}+126X^{5}+\cdots .}$
${\displaystyle AB=2X-6X^{2}+14X^{3}-26X^{4}+44X^{5}+\cdots .}$

${\displaystyle 44X^{5}=(1\times 6X^{5})+(5X^{2}\times 4X^{3})+(9X^{4}\times 2X).}$

${\displaystyle {\frac {1}{1+X}}=1-X+X^{2}-X^{3}+X^{4}-X^{5}+\cdots .}$

${\displaystyle [X^{5}]A=-11}$

## 形式幂级数的环结构

### 定义

${\displaystyle R[[X]]}$可以定义为${\displaystyle R}$上变量为${\displaystyle X}$的多项式环完备化（对于特定的度量）后得到的。这个定义自然就赋予了${\displaystyle R[[X]]}$以拓扑环的结构（同时也赋予了完备度量空间的结构）。不过空间完备化所需要的步骤过于繁琐，而建构${\displaystyle R[[X]]}$所需要的并没有那么多。以下将对${\displaystyle R[[X]]}$的环结构和拓扑结构分别定义，更为明晰，容易理解。

#### 环结构

${\displaystyle R^{\mathbb {N} }=\{(a_{n})_{n\in \mathbb {N} },\,\,\,\forall n\in \mathbb {N} ,\,a_{n}\in R\}.}$

${\displaystyle R^{\mathbb {N} }}$中的元素可以定义加法和乘法：

${\displaystyle (a_{n})_{n\in \mathbb {N} }+(b_{n})_{n\in \mathbb {N} }=\left(a_{n}+b_{n}\right)_{n\in \mathbb {N} }}$
${\displaystyle (a_{n})_{n\in \mathbb {N} }\times (b_{n})_{n\in \mathbb {N} }=\left(\sum _{k=0}^{n}a_{k}b_{n-k}\right)_{n\in \mathbb {N} }.}$

${\displaystyle x\in R\,\,\mapsto \,(x,0,0,...)}$

${\displaystyle (a_{0},a_{1},a_{2},\ldots ,a_{n},0,0,\ldots )\mapsto a_{0}+a_{1}X+\cdots +a_{n}X^{n}=\sum _{i=0}^{n}a_{i}X^{i}}$

${\displaystyle (a_{0},a_{1},a_{2},\ldots ,a_{n},\ldots )\mapsto \varphi (a_{0},a_{1},a_{2},\ldots ,a_{n},\ldots )=a_{0}+a_{1}X+\cdots +a_{n}X^{n}+\cdots =\sum _{i\in \mathbb {N} }a_{i}X^{i}}$

${\displaystyle \left(\sum _{i\in \mathbb {N} }a_{i}X^{i}\right)+\left(\sum _{i\in \mathbb {N} }b_{i}X^{i}\right)=\sum _{n\in \mathbb {N} }\left(a_{n}+b_{n}\right)X^{n}.}$

${\displaystyle \left(\sum _{i\in \mathbb {N} }a_{i}X^{i}\right)\times \left(\sum _{i\in \mathbb {N} }b_{i}X^{i}\right)=\sum _{n\in \mathbb {N} }\left(\sum _{k=0}^{n}a_{k}b_{n-k}\right)X^{n}.}$

#### 拓扑结构

${\displaystyle \varphi (a_{0},a_{1},a_{2},a_{3},\ldots )=\sum _{i=0}^{\infty }a_{i}X^{i},\qquad (1)}$

• 我们可以在${\displaystyle R}$上定义离散拓扑的结构，然后将${\displaystyle R^{\mathbb {N} }}$作为可数个${\displaystyle R}$积空间，将其上的拓扑定义为积拓扑
• 我们也可以直接在${\displaystyle R^{\mathbb {N} }}$上定义类似于p进数拓扑的${\displaystyle I}$进拓扑，其中的${\displaystyle I=(X)}$是环结构中由${\displaystyle X}$生成的理想，也就是由所有${\displaystyle \sum _{i=1}^{\infty }a_{i}X^{i}}$形式的形式幂级数构成的集合。
• 对不熟悉一般的点集拓扑学的读者，也可以建立一个具体的度量（也就是定义“距离”）来定义拓扑。比如定义两个数列${\displaystyle a=(a_{n})_{n\in \mathbb {N} }}$${\displaystyle b=(b_{n})_{n\in \mathbb {N} }}$的距离：
${\displaystyle d(a,b)={\begin{cases}2^{-\omega (a-b)}&\quad a-b\neq 0\\0&\quad a-b=0\end{cases}}}$

${\displaystyle s_{k}=(a_{0},a_{1},\ldots ,a_{k},0,0,\ldots )}$

${\displaystyle (a_{0},a_{1},a_{2},a_{3},\ldots )=\lim _{k\to \infty }s_{k}}$

${\displaystyle d'(\sum _{i=0}^{\infty }a_{i}X^{i},\sum _{i=0}^{\infty }b_{i}X^{i})={\begin{cases}2^{-\omega '},\,\,\omega '=\min _{n}\{a_{n}\neq b_{n}\}&\quad \exists a_{n}\neq b_{n}\\0&\quad \forall n,\,\,a_{n}=b_{n}\end{cases}}}$

${\displaystyle \sum _{i=0}^{\infty }a_{i}X^{i}=:\varphi (a_{0},a_{1},a_{2},a_{3},\ldots )=\lim _{k\to \infty }\varphi (s_{k})=\lim _{k\to \infty }\sum _{i=0}^{k}a_{i}X^{i}}$