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 ex at a = 0 is
출처 : http://en.wikipedia.org/wiki/Taylor_series
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