Example: The backward difference for $u'(t)$

Backward difference approximation to $u'$: $$\begin{equation} \lbrack D_t^- u\rbrack^n = \frac{u^{n} - u^{n-1}}{\Delta t} \approx u'(t_n) \tag{1} \tp \end{equation}$$

Define the truncation error of this approximation as $$\begin{equation} R^n = [D^-_tu]^n - u'(t_n)\tp \tag{2} \end{equation}$$

The common way of calculating $R^n$ is to

1. expand $u(t)$ in a Taylor series around the point where the derivative is evaluated, here $t_n$,
2. insert this Taylor series in (2), and
3. collect terms that cancel and simplify the expression.