Let $R$ be a commutative ring. I have seen that $c_0+c_1x+c_2x^2+\cdots.+ c_nx^n \in R[x]$ is nilpotent iff $c_i$ is nilpotent for all $i = 0,1, ... ,n$. But similar result is not true for $R[[x]]$. However, if we consider $R$ as a commutative ring with bounded index of nilpotency, then we get a similar result.
Recently I have seen a result of McCoy. It says that $f(x)\in R[x]$ is a zero divisor iff there exists a non-zero $r \in R$ s.t. $rf(x)=0$. This result is also not true if we replace $R[x]$ by $R[[x]]$. I want to know if we consider $R$ as a commutative ring with bounded index of nilpotency, then does the result become true for $R[[x]]$ also? Any hint/proof/counter example?