I have a quick question about multiplying formal series.
For a little notation, I write $\langle x^n\rangle F(x)$ for the $n$-th coefficient in a formal series $F(x)$. Also, a sequence $\{F_k(x)\}$ in the formal power series ring $R[[X]]$ converges to $G(x)$ if the sequence $\{\langle x^n\rangle F_k(x)\}$ converges to $\langle x^n\rangle G(x)$, in the discrete topology on the ring $R$.
I few days ago I saw that $\sum_{k=1}^\infty F_k(x)$ converges iff $\{F_k(x)\}$ converges to $0$.
When can one conclude that $\prod_{k=1}^\infty F_k(x)$ converges? My guess is that it happens iff $\{F_k(x)\}$ converges to $1$. Is this indeed the case, or does extra precaution need to be made when multiplying?
(P.S. In light of Riemann's rearrangement theorem for convergent series, is this product affected in any way by rearranging the terms? Just an extra curiosity, if there's time to answer.)
Thank you,