What is $k[x,y]_{(x,y)}$? After intensive internet research, I couldn't find any source that gave me a solution, probably the question is too easy. It would be great, if someone can tell me if I'm correct or not.
Let $k$ be any field (I'm usually working with $\mathbb{C}$, but every other field should be okay as well), is the polynomial ring localized at $(x,y)$ isomorphic to the formal power series ring, i.e.
$$
k[x,y]_{(x,y)} \simeq k[[x,y]]?
$$
Here $(x,y)$ is seen as an ideal, so the localization at $(x,y)$ could be interpreted as dividing by $k[x,y]\setminus (x,y)$.
Why should this be true? As a heuristic: On the LHS, only $x$ and $y$ (and multiples of these) have no inverse, the same is true for the RHS.
Does this hold for an arbitrary finite amount of variables? I would say yes.
Am I correct and it really is this easy or am I just standing on the hose?

No, they are not necessarily isomorphic. For one thing, if you choose $k$ to be countable, then $k[x,y]_{(x,y)}$ is countable, but $k[[x,y]]$ is uncountable.
For another thing, if I recall correctly, the former one is not complete with respect to its maximal ideal, whereas the second one is.
But they do superficially look similar: two local algebras whose maximal ideal is generated by $(x,y)$.
