The Sum May Not Converge
Our formula for the power series is
For certain values of x and ak, a power series can be infinite. Let's go back to the example we introduced earlier in this lesson
The sum of this series that tells us that the series only converges when |x| < 1 (by the divergence test). But in more general power series, there are three distinct possibilities that we can encounter.
Three Possibilities for Convergence
|Theorem: Only Three Convergence Results are Possible
|A general power series series
can only have three possibilities:
The constant R, if it exists, is called the radius of convergence. The interval of convergence of a power series, is the interval over which the series converges.