Taylor series

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.

A function can be approximated by using a finite number of terms of its Taylor series. The polynomial formed by taking some initial terms of the Taylor series is called a Taylor polynomial. The Taylor series of a function is the limit of that function's Taylor polynomials as the degree increases, provided that the limit exists.

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 in that interval.

Definition

The Taylor series of a real or complex-valued function $f(x)$ that is infinitely differentiable at a real or complex number $a$ is

$\sum_{n=0}^\infty \frac{f^{( n )}(a)}{n!}(x-a)^n$

The derivative of order zero of f is defined to be f itself and (x − a)⁰ and 0! are both defined to be 1. When a = 0, the series is also called a Maclaurin series.

Examples

The Taylor series for any polynomial is the polynomial itself.

$e^x=\sum_{n=0}^\infty \frac{x^n}{n!}=1+x+\frac{x^2}{2}+\frac{x^3}{6}+...$

$\frac{1}{1-x}=\sum_{n=0}^\infty x^n=1+x+x^2+x^3+...$

$ln(1-x)=-\sum_{n=1}^\infty \frac{x^n}{n}=-x-\frac{x^2}{2}-\frac{x^3}{3}-\frac{x^4}{4}-...$

$sin(x)=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}x^{2n+1}=x-\frac{x^3}{6}+\frac{x^5}{120}-...$

$cos(x)=\sum_{n=0}^\infty\frac{(-1)^n}{(2n)!}x^{2n}=\frac{x^2}{2}-\frac{x^4}{24}+\frac{x^6}{720}-...$

References