Determining if $A(B(x))$ is a formal power series I know that by a theorem, $A(B(x))$ is a formal power series if $b_0=0$.
I wasn't sure if it works the other way around.
Can I also say that if $b_0\ne0$, $A(B(x))$ is not a formal power series?
If it doesn't work the other way around, what other techniques can I perform to determine if $A(B(x))$ is a formal power series when $b_0\ne0$?
 A: The problem is that an infinite sum of nonzero elements of the coefficient ring (which you would need to do to compute the composition, unless $A$ had finitely many terms) is not defined without adding additional structure to the ring. Thus it is correct to say the composition is not necessarily a formal power series if $A$ has infinitely many terms (in particular plugging in a nonzero constant is invalid). 
A: There is a simple, natural notion of convergence that settles this and related matters, and it can be presented in a elementary way that requires no knowledge of topology. For example, see the discussion below (excerpted from Stanley's classic $ $ Enumerative Combinatorics I). $ $ In particular, the final paragraph shows how this applies to your problem of composition of formal power series.
Note $ $ Beware that there is widespread confusion between formal vs. analytic power series, and this often clouds discussion of this and related matters (as Rota often remarked, even some eminent mathematicians have published nonsense based on such confusion). For example, see this closely related question on products of formal power series (which was initially clouded by such confusion - see esp. the deleted answer and comments). It seems that - just as for polynomials - it proves difficult for some to shed  analytic (function) bias and pass to purely algebraic (formal) abstraction.



