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Taylor Series

Suppose that f(x) is a function, and that all the derivatives f ', f '', f ''', etc. exist at x = a. Then the Taylor Series of f(x) is the power series

 

or, in sigma notation,

A Maclaurin series is a Taylor series in the case where a = 0.

The partial sums of the Taylor series are called Taylor polynomials. These can be used to approximate the function in the neighborhood of x = a.