Let $(R, (\pi))$ be a discrete valuation ring with residue class field $R/(\pi) \cong k$. It is well known that if $k$ embedds into $R$, then there is an isomorphism of the completion $\hat{R} \cong k [[ T ]]$.
Assume $A$ is a $k$-algebra. How can I show that the (according) completion $\widehat{A \otimes_k R}$ (along the ideal $(1 \otimes \pi)$) is isomorphic to $A[[T]]$?
Use this result with $A=R$ and $B=R\otimes_kA$ and $\widehat{k[[T]]\otimes_kA}\simeq A[[T]]$.