Taylor Expansion – Taylor Series for Functions from R^n to R^n

Question

I've been told that we can write a taylor series for functions $f:R^n\rightarrow R$ but we can't write one for $f:R^n\rightarrow R^n$. I'm not quite sure why this not possible, but I suspect it have something to do with the mean value theorem. Could anyone shed some light on this?

Answer 1

If $f : \mathbb{R}^n \to \mathbb{R}^n$, then you can split $f = (f_1, f_2, \ldots, f_n)$, and (under the usual smoothness assumptions) each component can be expanded in a Taylor series.