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,

  • $\begingroup$ Hmm, you use the "two-sided" iff above; but I don't understand how the converse: "if { $\small F_k(x) $} converges to 0 then $\sum F_k(x) $ converges " , could be true - I simply consider that statement on the level of single coefficients only, and there it is not true. Or do I misunderstand something here? $\endgroup$ – Gottfried Helms Feb 6 '12 at 10:25

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.

  • $\begingroup$ Thanks for your answer Marc! Sorry if I'm learning things in a less than optimal order, I'm just trying to pick up math on my own time now. $\endgroup$ – Adelaide Dokras Feb 7 '12 at 5:34
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    $\begingroup$ @Adeal: Don't take that initial comment too personally. In almost all curricula numerical series are treated before formal power series, apparently because it is considered easier to study finite (sort of) objects with infinite operations on them (limits of sums) than to study infinite objects with finite operations on them (more precisely on each of their components). But I find the latter both easier and more fun. Besides, manipulating series formally comes about quite naturally; for instance Newton found his infinite binomial expansion this way. $\endgroup$ – Marc van Leeuwen Feb 9 '12 at 8:02

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