Let $R$ be a ring. On the ring of polynomials $A[X]$ I can define an evaluation map in $r \in R$ (the morphism that sends $X$ to $r$, extended by linearity and multiplication). This allows to "evaluate" polynomials at $r$. Of course, this cannot be replicated verbatim on arbitrary formal series rings, unless some notion of convergence in the base ring is used.
What puzzles me, though, is the following: for $R[X]$, the evaluation of $f \in R[X]$ at $r$ can also be viewed as the image of $f$ in $R[X] / (X-r)$. Why can't we do the same for $R[[X]]$? In other words, is there an explicit description of $R[[X]] / (X-r)$? If so, is it isomorphic to $R$? If so, is this isomorphism canonical, or is there a canonical embedding of this quotient into some space independent of $r$ , so that in particular one may compare the evaluations at different points (and do algebraic operations on them)?
As Daniel Gerigk has said in the comments, the element $X-r\in k[[X]]$ is invertible for $r\neq 0$. In this case, $k[[X]]/(X-r)=0$ so this version of evaluation is not very useful.