(i) A general multistep method for the numerical approximation to the scalar differential equation $y^{\prime}=f(t, y)$ is given by

$$\sum_{\ell=0}^{s} \rho_{\ell} y_{n+\ell}=h \sum_{\ell=0}^{s} \sigma_{\ell} f_{n+\ell}, \quad n=0,1, \ldots$$

where $f_{n+\ell}=f\left(t_{n+\ell}, y_{n+\ell}\right)$. Show that this method is of order $p \geqslant 1$ if and only if

$$\rho\left(\mathrm{e}^{z}\right)-z \sigma\left(\mathrm{e}^{z}\right)=\mathcal{O}\left(z^{p+1}\right) \quad \text { as } \quad z \rightarrow 0$$

where

$$\rho(w)=\sum_{\ell=0}^{s} \rho_{\ell} w^{\ell} \quad \text { and } \quad \sigma(w)=\sum_{\ell=0}^{s} \sigma_{\ell} w^{\ell}$$

(ii) A particular three-step implicit method is given by

$$y_{n+3}+(a-1) y_{n+1}-a y_{n}=h\left(f_{n+3}+\sum_{\ell=0}^{2} \sigma_{\ell} f_{n+\ell}\right)$$

where the $\sigma_{\ell}$ are chosen to make the method third order. [The $\sigma_{\ell}$ need not be found.] For what values of $a$ is the method convergent?