Taylor series

American  

noun

Mathematics.

1. an approximation of a given function f at a particular point x, in terms of values of the function and its derivatives at a neighboring point x 0 , by a power series in which the terms are given by f (n) (x0 ) (x−x0 ) n/n !, where f (n) (x0 ) is the derivative of order n evaluated at point x 0 .

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Etymology

Origin of Taylor series

1905–10; after Brook Taylor

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• Maclaurin series vs. Taylor series

Example Sentences

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Taylor series

In the preceding section, we defined Taylor series and showed how to find the Taylor series for several common functions by explicitly calculating the coefficients of the Taylor polynomials.

From Textbooks ● Mar. 30, 2016

In general, Taylor series are useful because they allow us to represent known functions using polynomials, thus providing us a tool for approximating function values and estimating complicated integrals.

From Textbooks ● Mar. 30, 2016

We now consider the more general question: if a Taylor series for a function f converges on some interval, how can we determine if it actually converges to f ?

From Textbooks ● Mar. 30, 2016

This theorem allows us to bound the error when using a Taylor polynomial to approximate a function value, and will be important in proving that a Taylor series for f converges to f .

From Textbooks ● Mar. 30, 2016

In this example, we differentiated a known Taylor series to construct a Taylor series for another function.

From Textbooks ● Mar. 30, 2016