This the highest voted question for tag "series-expansion" -- https://mathematica.stackexchange.com/questions/15023/multivariable-taylor-expansion-does-not-work-as-expected

I just got tripped by this after not reading the docs carefully, a bit unexpected to have a built-in Series functionality for sequential differentiation but not for multivariate differentiation.

Perhaps multivariate Taylor could be specified as nested list:

Series[f[a,b], {{a, b}, {a0, b0}, 1}]

Example

Clear[A, B]; multiTaylor[f_, {vars_?VectorQ, pt_?VectorQ, n_Integer?NonNegative}] :=    Sum[Nest[(vars - pt) . # &, (D[f, {vars, \[FormalK]}] /. 
       Thread[vars -> pt]), \[FormalK]]/\[FormalK]!, {\[FormalK], 0, 
    n}, Method -> "Procedural"];
multiTaylor[a/b, {{a, b}, {a0, b0}, 1}] // FullSimplify (* expected result *)

Series[a/b, {a, a0, 1}, {b, b0, 1}] // Normal // FullSimplify (* unexpected result *)