(a) An $s$-step method for solving the ordinary differential equation

$$\frac{d \mathbf{y}}{d t}=\mathbf{f}(t, \mathbf{y})$$

is given by

$$\sum_{l=0}^{s} \rho_{l} \mathbf{y}_{n+l}=h \sum_{l=0}^{s} \sigma_{l} \mathbf{f}\left(t_{n+l}, \mathbf{y}_{n+l}\right), \quad n=0,1, \ldots$$

where $\rho_{l}$ and $\sigma_{l}(l=0,1, \ldots, s)$ are constant coefficients, with $\rho_{s}=1$, and $h$ is the time-step. Prove that the method is of order $p \geqslant 1$ if and only if

$$\rho\left(e^{z}\right)-z \sigma\left(e^{z}\right)=O\left(z^{p+1}\right)$$

as $z \rightarrow 0$, where

$$\rho(w)=\sum_{l=0}^{s} \rho_{l} w^{l}, \quad \sigma(w)=\sum_{l=0}^{s} \sigma_{l} w^{l}$$

(b) Show that the Adams-Moulton method

$$\mathbf{y}_{n+2}=\mathbf{y}_{n+1}+\frac{h}{12}\left(5 \mathbf{f}\left(t_{n+2}, \mathbf{y}_{n+2}\right)+8 \mathbf{f}\left(t_{n+1}, \mathbf{y}_{n+1}\right)-\mathbf{f}\left(t_{n}, \mathbf{y}_{n}\right)\right)$$

is of third order and convergent.

[You may assume the Dahlquist equivalence theorem if you state it clearly.]