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.

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

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

$\left(t-\frac{1}{6}{t}^3+\frac{1}{120}{t}^5+O\left(t^7\right)\right)e_x\+\left(1+t+\frac{1}{2}{t}^2+\frac{1}{6}{t}^3+\frac{1}{24}{t}^4+\frac{1}{120}{t}^5+O\left(t^6\right)\right)e_y$

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

$\left(1-\frac{1}{6}{t}^2+\frac{1}{120}{t}^4+O\left(t^6\right)\right)e_x\+\left(\frac{1}{2}t-\frac{1}{24}{t}^3+O\left(t^5\right)\right)e_y$

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

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