Taylor series

A Taylor series is a method for representing a function in a neighborhood of a single point. It is an infinite series of terms that approximate the function...

A Taylor series is a method for representing a function in a neighborhood of a single point. It is an infinite series of terms that approximate the function locally.

The Taylor series is defined by the following general formula:

$f(z) = \sum_{n=0}^{\infty} a_n (z - a)^n$

where:

(f(z)) is the function we want to represent with the Taylor series.
(a_n) are the coefficients of the Taylor series.
(z) is the variable.

The Taylor series is centered at (a), and the terms are calculated by multiplying the corresponding coefficient by the distance from (a) to the point in the domain.

Here are some Taylor series for common functions:

sin(x): $\sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac{(x-a)^n}{n!}$
cos(x): $\cos(x) = \sum_{n=0}^{\infty} \frac{(x-a)^n}{n!}$
e: $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$

Taylor series have several important properties:

They are continuous functions of the variable.
They converge to the function in the neighborhood of the center.
The rate of convergence can be determined by the ratio of consecutive coefficients.

Taylor series are widely used in various engineering and scientific applications, such as solving differential equations, approximating function values, and analyzing physical systems