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,
Best Answer
A preliminary comment: if you had studied formal power series before studying "normal" power series (as I would advise anyone to do), you probably would never have had any difficulty with this.
Saying that $\sum_{k=1}^\infty F_k(X)$ converges means that for every $n\in\mathbf N$ the coefficient $\langle X^n\rangle \sum_{k=1}^\infty F_k(X)$ converges in the discrete topology on $R$. This means that this coefficient becomes stable after some finite number $N(n)$ of terms have been accumulated into the partial sum, in other words none of the terms $F_k(X)$ for $k>N(n)$ contribute anything to it. Since its contribution is $\langle X^n\rangle F_k(X)$, this means that $\langle X^n\rangle F_k(X)=0$ whenever $k>N(n)$, and since this is true for every $n$, one has $\lim_{k\to\infty}F_k(X)=\sum_{n\in\mathbf N}0X^n=0$ by definition of convergence in $R[[X]]$.
Since rearranging terms will not alter whether a particular coefficient ultimately becomes zero, nor (if it does) the sum of the nonzero coefficients, convergence and limits are unaffected by rearranging terms.