Taylor Series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor in 1715. If the Taylor series is centered at zero, then that series is also called a Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.

It is common practice to approximate a function by using a finite number of terms of its Taylor series. Taylor's theorem gives quantitative estimates on the error in this approximation. Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial. The Taylor series of a function is the limit of that function's Taylor polynomials, provided that the limit exists. A function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an open interval (or a disc in the complex plane) is known as an analytic function.

Read more about Taylor Series:  Definition, Examples, History, Analytic Functions, Approximation and Convergence, List of Maclaurin Series of Some Common Functions, Calculation of Taylor Series, Taylor Series As Definitions, Taylor Series in Several Variables, Fractional Taylor Series

Famous quotes containing the words taylor and/or series:

    Oh, what a might is this whose single frown
    Doth shake the world as it would shake it down?
    Which all from nothing fet, from nothing all;
    Hath all on nothing set, lets nothing fall.
    Gave all to nothing man indeed, whereby
    Through nothing man all might Him glorify.
    —Edward Taylor (1645–1729)

    Galileo, with an operaglass, discovered a more splendid series of celestial phenomena than anyone since.
    Ralph Waldo Emerson (1803–1882)