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2.2.1 Taylor Series Methods

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.