  We introduce general methods to analyse the Goodwillie tower of the identity
functor on a wedge $X \vee Y$ of spaces (using the Hilton-Milnor theorem) and
on the cofibre $\mathrm{cof}(f)$ of a map $f: X \rightarrow Y$. We deduce some
consequences for $v_n$-periodic homotopy groups: whereas the Goodwillie tower
is finite and converges in periodic homotopy when evaluated on spheres
(Arone-Mahowald), we show that neither of these statements remains true for
wedges and Moore spaces.
