# Order of a linear multistep method

Given a linear multistep method $\sum_{j=0}^{k}\alpha_jy_{n+j}=h\sum_{j=0}^k\beta_jf_{n+j}$ how to show using Taylor series expansion that the method is of order $p$ if and only if $D_0=D_1=D_2=\cdots=D_p=0$, where

$D_0=\alpha_0+\alpha_1+\cdots+\alpha_k$, $D_1=-t\alpha_0+(1-t)\alpha_1+(2-t)\alpha_2+\cdots+(k-t)\alpha_k-(\beta_0+\beta_1+\cdots+\beta_k)$, . . . $D_q=\frac{1}{q!}[(-t)^q\alpha_0+(1-t)^q\alpha_1+(2-t)^q\alpha_2+\cdots+(k-t)^q\alpha_k]-\frac{1}{(q-1)!}[(-t)^{q-1}\beta_0+(1-t)^{q-1}\beta_1+\cdots+(k-t)^{q-1}\beta_k]$ for $q=2, 3, \ldots$.