The series(f, v) calling sequence maps the top-level series function over the components of the Vector f.  If f is not a Vector, the arguments are passed to the top-level series function.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{VectorCalculus}\right]\right)\:$

$\mathrm{series}\left(⟨\sin \left(t\right)\,\exp \left(t\right)⟩\,t=0\right)$

$\mathrm{series}\left(⟨\frac{\sin \left(t\right)}{t}\,\frac{1-\cos \left(t\right)}{t}⟩\,t=0\right)$

$\mathrm{series}\left(\log \left(1+z\right)\,z\right)$

$z-\frac{1}{2}{z}^2+\frac{1}{3}{z}^3-\frac{1}{4}{z}^4+\frac{1}{5}{z}^5+O\left(z^6\right)$