The Radius of Convergence Theorem
For a given power series,precisely one of the following may happen

the series converges only for

the series converges for all

there is some real numbersuch thatconverges and converges absolutely ifanddiverges if
Example: find the radius of convergence of the sequence
in this example. The ratio test demandsfor convergence so
Assoand asthe right hand side tends to 0 so the sequence converges only forThe radius of convergence is 0.
Example: find the radius of convergence of the sequence
The ratio test demandsfor convergence so
Asfor allso the sequence converges for all
Example: find the radius of convergence of the sequence
The ratio test demandsfor convergence so
The radius of convergence is 1.