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.

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2. Discussion [Using Flash] [Using Java]

3. Discussion [Using Flash] [Using Java]

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 Theorem. Given the domain of F is either {c} the interval (c - R, c + R) for some R > 0 together possibly with one or both of the endpoints; or 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]