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.

- 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 .

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

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