Let $R$ be a ring. The ring of formal power series $R[[X]]$ is the ring $\fun(\N, R)$ with addition and multiplication defined by
\[\begin{align*} a + b \enspace&:\enspace n \mapsto a_n + b_n, \\ a \cdot b \enspace&:\enspace n \mapsto\sum_{k = 0}^n a_k b_{n - k} \end{align*}\]for all $a$, $b \in R[[X]]$. $X$ is merely a formal symbol, called an indeterminate.
For each $a \in R[[X]]$, we write
\[a = \sum_{k = 0}^\infty a_k X^k.\]Each $a_k$ ($k \in \N$) is called a coefficient of $a$.