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.