Definition. A series of the form

where x is a variable, where {an} is a sequence and c is a constant, is called a power series about c (or centered at c).

We can define a function

Problem. What is the domain of F?

In particular, the domain of F is the collection of those values of x for which the series is convergent. We note that c is in the domain of F.

Examples.


  1. [Discussion.]


  2. Discussion [Using Flash] [Using Java]


  3. Discussion [Using Flash] [Using Java]


  4. Discussion [Using Flash] [Using Java]

Theorem. Given

the domain of F is either

  1. {c}

  2. the interval (c - R, c + R) for some R > 0 together possibly with one or both of the endpoints; or

  3. the collection of all real numbers.

In case 2, R is called the radius of convergence. In case 1 and case 3, we say that the radius of convergence is 0 and ¥, respectively.

The domain of F is generally referred to as the interval of convergence.

Examples.


  1. Discussion [Using Flash] [Using Java]


  2. Discussion [Using Flash] [Using Java]