Power series.

Definition 10.3.1   A power series is a series of the form

where is a real variable, is a fixed real number called the center.

The sequence of real numbers is called the sequence of coefficients of the series.

Example 10.3.2

• The power series is a geometric series with ratio equal to . It is convergent for and its sum is .
• The power series is a geometric series with ratio equal to . It is convergent for , i.e. for and its sum is .

Example 10.3.3   Consider the series . we use the ratio test (v.s. 3.7) in order to determine the domain of convergence of the series. Here we have:

Thus, , for any real . It follows that the given power series is convergent for any real (its sum is , v.i. 5.2).

Example 10.3.4   Consider the series . we use the ratio test (v.s. 3.7) in order to determine the domain of convergence of the series. Here we have:

Thus, , for any real . It follows that the given power series is convergent for any real such that .

Proposition 10.3.5   If the power series is convergent for , then it is absolutely convergent for any such that .

If this power series is divergent for , then it is divergent for any such that .

It follows that the domain of convergence of a power series is either the whole of or an interval centered at , i.e. an interval of the form (maybe closed or open-closed), wherevthe number v is called the convergence radius. In the first case, the radius of convergence is said to be infinite. .

An algorithm for testing the convergence of a power series:

Noah Dana-Picard 2007-12-28