Set $A$ as a commutative noetherian local ring with maximal ideal $\mathfrak{m}$ generated by $m_1,\ldots,m_n$.
I believe, every element $a \in A$ can be written as a finite sum: \begin{align*} a= \sum_{k_1,\ldots,k_n \geq 0} a_{k_1,\ldots,k_n}m_1^{k_1}\ldots m_n^{k_n} \end{align*} with $a_{k_1,\ldots,k_n}\in A \setminus \mathfrak{m}$.
Am I wrong?
EDIT:
and there fore every element $\alpha \in A[[X]]$ can be written as (perhaps not infinite sum): \begin{align*} \alpha= \sum_{k_1,\ldots,k_n,k_{n+1} \geq 0} a_{k_1,\ldots,k_n,k_{n+1}}m_1^{k_1}\ldots m_n^{k_n}X^{k_{n+1}} \end{align*} with $a_{k_1,\ldots,k_n,k_{n+1}} \in A\setminus \mathfrak{m}$
When $a\in \frak{m}$, it is trivial. And when $a\notin \frak{m}$, we define $a_{0,\dots,0}:=a$ and otherwise $a_{k_1\dots k_n}=0$, then $a=a_{0,\dots,0}m_1^{0}\cdots m_n^{0}$.
EDIT:
That is also true. First we write $\alpha=\sum_{k_{n+1}\ge 0}\alpha_{k_{n+1}}X^{k_{n+1}}$, and we apply the above fact for each $\alpha_{k_{n+1}}\in A$.