Jacobson radical of formal power series over an integral domain

Consider the (surjective) ring homomorphism $\varepsilon\colon R[[x]]\to R$ that sends an element $f\in R[[x]]$ to its constant term. If $I$ is an ideal of $R[[x]]$, then $\varepsilon(I)$ is an ideal of $R$ and we have the induced surjective homomorphism $$ \varepsilon_I\colon R[[x]]/I\to R/\varepsilon(I),\qquad \varepsilon_I(f+I)=\varepsilon(f)+\varepsilon(I) $$ If $I$ is maximal, then $R[[x]]/I$ is a field, so $\varepsilon_I$ is an isomorphism. In particular, $f\in I$ if and only if $\varepsilon(f)\in \varepsilon(I)$ and $\varepsilon(I)$ is a maximal ideal.

Conversely, if $J$ is a maximal ideal of $R$, then $I=J+xR[[x]]$ is a maximal ideal of $R[[x]]$ and $J=\varepsilon(I)$ (prove it).

Thus $f$ belongs to every maximal ideals of $R[[x]]$ if and only if its constant term belongs to every maximal ideal of $R$, that is, to the Jacobson radical $J(R)$. Therefore $$ J(R[[x]])=J(R)+xR[[x]] $$