Talk:Formal power series
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I've slightly improved (in my opinion) the sentence on k!ak in the section on differentiation. However, I'd probably prefer k! to be regarded as a natural number, and to view this term as module multiplication of the additive group of R regarded as a module over the integers. I'm not sure how to state that concisely. How about
- here nak = ak+...+ak (n summands)
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Formal power series as functions
I think this article is quite well written, although I don't know if the introduction of the metric d() is standard and/or necessary for the notation <math>\sum a_kX^K<math>, which I was taught to be a mere notation for the sequence <math>(a_k)<math>.
I like this approach, but it may lead to confusion. Indeed, this metric is the only one introduced on this page, while the section "Formal power series as functions", starting with
In mathematical analysis, every convergent power series defines a function with values in the real or complex numbers.
clearly does not refer to this metric (for "convergent"), but to the topology in C (or in some complex function space), for the series of partial sums (of numbers or functions?) associated to the power series in the "obvious" way (but a precise define is not so immediate at all...).
As it stands, this phrase is meaningless, because any power series is convergent according to what precedes, even if it has convergence radius of zero. So I think we should change it, so that this subsection becomes as "perfect" as the rest of the article.
If the author of this article can find a more precise introduction for this subsection in the same style as what precedes, I would appreciate. (I don't know what is the most "gentle" way to go from a power series to the associated analytic function.)