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I'm aware of Taylor's theorem for polynomials over $\mathbb{R}$. More generally though, if working with formal power series over a coefficient ring which contains $\mathbb{Q}$, why does Taylor's formula still hold?

Thank you.

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    What exactly is the Taylor's formula you are alluding to in the context of formal power series?2012-01-13
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    I’m guessing that you mean [Newton’s series](http://en.wikipedia.org/wiki/Newton_series#Newton_series), $$f(x)=\sum_{k=0}^\infty\frac{\Delta^k[f](a)}{k!}(x-a)^{\underline k}=\sum_{k=0}^\infty\binom{x-a}k\Delta^k[f](a)\;.$$ In what sense do you mean *holds*? You might find the discussion in Graham, Knuth, & Patashnik, *Concrete Mathematics*, pp. 189-192, helpful.2012-01-13
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    @BrianM.Scott I mean holds as in, "is it true"? This is just the "formal derivative" and I'll always take $a=0$ as I'm interested in ordinary generating functions (with just $x^n$'s). The equality you posted makes sense in this context, but is it true?2016-09-23

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