Let $\mathbb{k}$ be a field and $A$ be a commutative $\mathbb{k}$-algebra. It is clear that when $A$ is an affine $\mathbb{k}$-algebra (that is, $A$ is finitely generated as a $\mathbb{k}$-algebra), the polynomial algebra $A[x]$ is affine both as $\mathbb{k}[x]$-algebra and $\mathbb{k}$-algebra. So it is natural to ask if $A[[x]]$ is affine as a $\mathbb{k}[[x]]$-algebra when $A$ is affine? (I know that $A[[x]]$ is not affine as a $\mathbb{k}$-algebra in general).
Intuition says no, but I have no idea how to prove or disprove it. After some thinking, I found that suppose $A$ can be generated by $\{a_{1},...,a_{m}\}$, then $A[[x]]$ may not be generated by $\{a_{1},...,a_{m}\}$ as a $\mathbb{k}[[x]]$-algebra in general. For example, consider the polynomial algebra $A=\mathbb{k}[y]$, which can be generated by $y$ . It is clear that $\mathbb{k}[[x]][y]\subsetneq \mathbb{k}[y][[x]]=A[[x]]$. And now I feel stuck.
Can you give me a hint or something more? Thanks!
Suppose that $k$ is any ring and $V$ is $k$-module that is not finitely generated. Then I claim that $V[[x]]$ is not countably generated as a $k[[x]]$-module. In particular, if $V$ is additionally a $k$-algebra, this implies $V[[x]]$ is also not countably generated as a $k[[x]]$-algebra.
To prove this, take any countable sequence of elements $(f_n)$ in $V[[x]]$; we will show it does not generate $V[[x]]$ as a $k[[x]]$-module. Since $V$ is not finitely generated, for each $n$ we can pick an element $a_n\in V$ which is not in the submodule generated by any of the coefficients of $f_0,\dots,f_n$ for powers of $x$ less than or equal to $n$. Now consider the element $a=\sum a_nx^n\in V[[x]]$. If $a$ were a $k[[x]]$-linear combination of $f_0,\dots,f_n$, then for each $m$, $a_m$ would be a $k$-linear combination of the coefficients of $f_0,\dots,f_n$ for powers of $x$ less than or equal to $m$. But by construction, this is false for $m=n$. Thus $a$ cannot be a $k[[x]]$-linear combination of $f_0,\dots,f_n$ for any $n$, so it is not in the submodule generated by the entire sequence $(f_n)$.