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 $a$.