The radius of convergence:

Let be a power series, and then the power series converges absolutely if and diverges if

The interval of convergence:

Every power series has a radius of convergence. The interval is the interval of convergence for a power series.

Maclaurin Series:

The Taylor series about is called the Maclaurin Series, i.e. for every

The Lagrange form of the Remainder:

If exists and is continuous on I then for any x in I, there exists a point between c and x such that

Taylor polynomials are discussed with the help of power series and the radius of convergence principle. Power series are important to the development of the radius and the interval of convergence ideas. Examples are given to show the radius and the interval of convergence principles.

The Taylor Series idea is created from the differentiation of a particular function a countable number of times. Polynomials are approximated using Taylor series and Maclaurin Series. The error of series is given with the use of Lagrange multipliers and how a new series is created from a given series.

Specific Tutorial Features:

• Step by step detailed examples showing the radius of convergence and interval of convergence principles.
  
• Taylor series are given with the use of theorems and example problems.
  
• Step by step explanation of Maclaurin Series and the error bound of series approximations.

Series Features:

• Concept map showing inter-connections of new concepts in this tutorial and those previously introduced.
  
• Definition slides introduce terms as they are needed.
  
• Visual representation of concepts
  
• Animated examples—worked out step by step
  
• A concise summary is given at the conclusion of the tutorial.