## The Taylor Series

A series known as the Taylor series is frequently used in the evaluation of functions by numerical methods. The Taylor series for the evaluation of a function F at the point x + h, given the value of the function and its derivatives at the point x, is

Fk(x)hk

k=i where Fk(.xj is the Ath derivative of the function at the point x, and q is the remainder or error term. As has been illustrated by examples we have seen earlier, the magnitude of £ decreases as k (the number of terms) increases.

To obtain a result that closely approximates the true value of a function, we need to sum a number of terms. Clearly, we will not have available to us (without a lot of work) values of a large number of derivatives of the function F, up to the Ath derivative. Fortunately, we will usually need only the first derivative, the first and second derivatives, or the first, second and third derivatives to obtain results of sufficient accuracy. We will use the Taylor series expansion of a function in several of the subsequent chapters.

The order of the approximation is determined by the highest-derivative term that is included in the approximation; thus the first-order Taylor series approximation is

the second-order approximation is and the third-order approximation is

F(x + F(x) + hF'(x) + — F"(x) + — F"'(x) (4-6)

Obviously, the accuracy of the approximation increases as the number of terms is increased. It is also obvious that the accuracy of the approximation will increase as h is made smaller. Higher-order terms will become more important as h is increased, or if the function is nonlinear.

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