Is an $n$-order multiple Taylor series equivalent to $n$ successive one-variable series expansions?

Question

I am dealing with a complicated function $F= F(\epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4)$ where $\epsilon_i \, , i \in \{1,2,3,4\}$ are small parameters. I have noticed that a multiple Taylor expansion leads to a drastically different result then taking 4 times a standard Taylor expansion for a single variable. I was wondering whether this is true always, or whether there exist situations where both expansions lead to identical results. Your help is highly appreciated and desirable.

Sincerely, Hartmut

Answer 1

Taylor series become complex when multiple parameters.

Just suppose that you use the first order : it means that you look for the tangent plane to the curve. Then, with two variables $x,y$ the bilinear model looks like $$F=a+b x+ cy +d xy$$ More parameters make life more difficult, for sure.

For three variables $x,y,z$, it would be $$F=a+b x+ cy +dz +exy+f xz +g yz + h xy z$$