Algebra
《Algebra》 · 第 154 页 · Chapter III，Section 5，Proposition 5.8 及其后命名段
Let $R$ be a ring and denote by $R[[x]]$ the set of all sequences of elements of $R$ $(a_0,a_1,\ldots)$. (i) $R[[x]]$ is a ring with addition and multiplication defined by: $(a_0,a_1,\ldots)+(b_0,b_1,\ldots)=(a_0+b_0,a_1+b_1,\ldots)$ and $(a_0,a_1,\ldots)(b_0,b_1,\ldots)=(c_0,c_1,\ldots)$, where $c_n=\sum_{j=0}^n a_jb_{n-j}=\sum_{k+j=n}a_kb_j$. (ii) The polynomial ring $R[x]$ is a subring of $R[[x]]$. (iii) If $R$ is commutative [resp. a ring with identity or a ring with no zero divisors or an integral domain], then so is $R[[x]]$. […] The ring $R[[x]]$ of Proposition 5.8 is called the ring of formal power series over the ring $R$. Its elements are called power series.