I have to solve the following problem regarding formal power series. I was wondering if anyone could show me a way to approach this problem. I am tempted to use techniques of convergence from analysis, but I don't know if this is waht I should do, since we are working with formal power series. Thanks for any help!
Recall that a sequence of formal power series $F_{k}(x) = \sum _{n} a^{(k)}_{n} x^{n}$ over a ring $R$ converges to $G(x) = \sum _{n} b_{n} x^{n}$ if for each $n$ the sequence $(a^{(1)}_{n},a^{(2)}_{n},\ldots)$ converges to $b_{n}$ in the discrete topology, i.e., if $a^{(k)}_{n} = b_{n}$ for all sufficiently large $k$.
Prove that if the factors $F_{k}(x)$ converge to $1$, then the product $\prod _{k=1}^{\infty }F_{k}(x)$ converges, and its value does not depend on the order of the factors. Prove also that if the constant term $a_{0}^{(k)}$ of $F_{k}(x)$ is a non-zero-divisor in $R$ for every $k$, then $\prod _{k=1}^{\infty }F_{k}(x)$ converges if and only if the $F_{k}(x)$ converge to $1$.