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Taylor sequences

The most straightforward way to generate an easily integrated approximation for is to take a Taylor series expansion of it,[2][pages 240-241] terminating the expansion after some predetermined number of terms.

The error due to terminating the series is proportional to the first term dropped. Terms of higher order will contribute less than lower order terms.

When is used directly in place of (that is, only the first term is kept), the technique becomes the first order Euler method.

Typically, a higher order approximation will result in a more accurate approximation, however to use a Taylor sequence of order , requires that the function be times differentiable. Looking at the force function in equation , one can see that the function is not easily differentiable. It is, in fact, two valued, due to the absolute value operation required to define the direction vector. Further, for very small distances, the denominator may become zero (due to underflow), and cause a singularity.