A Taylor series is a method of approximating a function using an infinite sum of terms which are expressed in terms of the functions derivatives at a point. The sum is evaluated at some point to approximate . It is a type of power series and thus has a radius of convergence

and it's Taylor series approximations at using , , , , and term(s).

The formula used to calculate the Taylor series approximation of a given function near a point is defined as follows:

• Definitions
  • denotes the factorial of
  • denotes the derivative of at the point

This could also be expressed as a sum of infinite terms

Typically though Taylor series are only evaluated up to a set amount of terms depending on the use case.

Uses

• Determining higher order derivatives at a point.md)
• Evaluate geometric series with an additional term.md)

Example

Find the Taylor series for around and determine it's radius of convergence.

Using the ratio test:

Hence since the series converges for all values of , the radius of convergence is .