Power Series
DEFINITION: If are constants and is a variable, then a series of the form



is called a power series in and a series of the form



is called a power series in

EXAMPLES:

1.

2.

3.

4.

5.

THEOREM: For any power series exactly one of the following is true:

(a) The series converges only for

(b) The series converges absolutely (and hence converges) for all real values of

(c) The series converges absolutely (and hence converges) for all in some finite open interval and diverges if or At either of the values or the series may converge absolutely, converge conditionally, or diverge, depending on the particular series.

THEOREM: For a power series exactly one of the following is true:

(a) The series converges only for

(b) The series converges absolutely (and hence converges) for all real values of

(c) The series converges absolutely (and hence converges) for all in some finite open interval and diverges if or At either of the values or the series may converge absolutely, converge conditionally, or diverge, depending on the particular series.

EXAMPLE: Find the interval of convergence of the series


EXAMPLE: Find the interval of convergence of the series


EXAMPLE: Find the interval of convergence of the series


EXAMPLE: Find the interval of convergence of the series


EXAMPLE: Find the interval of convergence of the series


EXAMPLE: Find the interval of convergence of the series


EXAMPLE: Find the interval of convergence of the series