# Analog of Taylor's formula for formal power series?

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? –  Jonas Meyer Jan 13 '12 at 7:52
I’m guessing that you mean Newton’s 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. –  Brian M. Scott Jan 13 '12 at 8:53