Ever since reading about "normal" power series, I've been interested in how formal power series work.
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 $R$.
One property I read is that the sum $\sum_{k=1}^\infty F_k(x)$ converges iff $\{F_k(x)\}$ converges to $0$. This kind of makes sense to me when thinking of power series, in that a power series diverges if its terms do not approach $0$.
Intuition aside, what is the more rigorous reason why this property is in fact true? Does rearranging the terms change the convergence or the limit to which it converges?
Thank you,