In textbooks these polynomials are not normally presented as an infinite series (the single variables are). What is the reason for this and are they equally allowed to be in infinite series form hence infinite order just like the single variable Taylor Polynomials? Or are there more issues about convergence to worry about in the multivarible case so they are not usually written in the infinite form? Also I've seen two versions of Taylor's theorem. One has f(x+h) in http://en.wikipedia.org/wiki/Multi-index under Taylor series: for an analytic function and the other f(x) in http://en.wikipedia.org/wiki/Taylor's_Theorem under Taylor's theorem for several variables. This is the one I was referring to in the above question. How are the two reunited? I think f(x+h) is more complete because it allows f(x) plus added terms to account for the function at a position x+h.