TITLE: A non-nilpotent formal power series with nilpotent coefficients
QUESTION [17 upvotes]: Does anyone have an example of a formal power series $$p=a_0+a_1x+ a_2x^2 + \cdots \in R[[x]]$$ ($R$ is a commutative ring) all of whose coefficients $a_i$ are nilpotent in $R$ such that $p$ is not nilpotent in $R[[x]]$? 

I know that if $p$ is nilpotent in $R[[x]]$ then all $a_i$'s are necessarily nilpotent in $R$, but I can't figure out a simple example that shows that the opposite is not true in general. Any help is appreciated.

REPLY [7 votes]: For an example, take
$$
R=\mathbb{Q}[t,t^{1/2},t^{1/3},\ldots]/(t)
$$
Form the power series $p(x)=\sum_{n\geq 1} a_nx^n$, with $a_n=t^{1/n}$.  I leave it to you to check that $p(x)$ is not nilpotent.