# Taylor series

### From Conservapedia

This article/section deals with mathematical concepts appropriate for a student in late high school or early university. |

The **Taylor series** of a function is useful for approximating a mathematical function near to some particular point. For a function *f*(*x*), the Taylor series about the point *x*_{0} is

where each of the derivatives is to be evaluated at *x* = *x*_{0}. If as the series converges, then it is exact. Otherwise, it can be used as an approximation. Often, Taylor series are performed around *x*_{0} = 0, in which case they are sometimes also known as a Maclaurin series.

### Examples of common Taylor series

#### Extensions of the Exponential Function

Consider the exponential of imaginary number *y**i*,

- = cos
*y*+*i*sin*y*

By the power laws then all complex numbers have an exponential,

*e*^{x + yi} = *e*^{x}(cos*y* + *i*sin*y*).