Perhaps the simplest one-step methods of order p are based on Taylor series expansion of the solution . If is continuous on , then Taylor's formula gives
where . The continuity of implies that it is bounded on and so
Using the fact that , (24) can be written in the form
where the total derivatives of f are defined recursively by
Comparison of (22) and (26) shows that to obtain a method of order p, we can let
This choice leads to a family of methods known as the Taylor series methods, given in the following algorithm.