Does $\mathbb Z_p \otimes \mathbb Z[[x]]=\mathbb Z_p[[x]]$ hold?
In particular, I don't know how to express $\sum\frac 1{q^n}x^n$ for $q\ne p$ as an element of the tensor product.
Or it should be $\mathbb Z_p \hat\otimes \mathbb Z[[x]]=\mathbb Z_p[[x]]$?