Taylor series - 테일러 급수전개

In mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It may be regarded as the limit of the Taylor polynomials. Taylor series are named after English mathematician Brook Taylor. If the series is centered at zero, the series is also called a Maclaurin series, named after Scottish mathematician Colin Maclaurin.


Definition

The Taylor series of a real or complex function ƒ(x) that is infinitely differentiable in a neighborhood of a real or complex number a, is the power series

which in a more compact form can be written as

where n! is the factorial of n and ƒ ⁽ⁿ⁾(a) denotes the nth derivative of ƒ evaluated at the point a; the zeroth derivative of ƒ is defined to be ƒ itself and (x − a)⁰ and 0! are both defined to be 1.

In the particular case where a = 0, the series is also called a Maclaurin series.

Examples

The Maclaurin series for any polynomial is the polynomial itself.
The Maclaurin series for (1 − x)⁻¹ is the geometric series

so the Taylor series for x⁻¹ at a = 1 is

By integrating the above Maclaurin series we find the Maclaurin series for −log(1 − x), where log denotes the natural logarithm:

and the corresponding Taylor series for log(x) at a = 1 is

The Taylor series for the exponential function e^x at a = 0 is

The above expansion holds because the derivative of e^x is also e^x and e⁰ equals 1. This leaves the terms (x − 0)ⁿ in the numerator and n! in the denominator for each term in the infinite sum.

출처 : http://en.wikipedia.org/wiki/Taylor_series



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