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.
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.
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.