The $v_n$-periodic Goodwillie tower on Wedges and Cofibres

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.