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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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    @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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A certain formal Taylor's theorem comes up fairly often in the theory of Vertex Algebras. Haisheng Li and James Lepowsky's introduction to vertex algebras spends a whole chapter on "formal calculus" proving (among many other things) a formal Taylor's theorem.

There is a more general formal Taylor theorem (taking into account formal logarithms) in HLZ (part II in a series of papers on logarithmic intertwining operators). A student of James Lepowsky named Thomas Robinson has written a bunch of papers refining various techniques of formal calculus. In particular this paper of Robinson has a fairly general Taylor theorem appearing as Theorem 4.1.