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 number
such that
converges and converges absolutely if
anddiverges if
Example: find the radius of convergence of the sequence
in this example. The ratio test demands
for convergence so
As
so
and as
the right hand side tends to 0 so the sequence converges only for
The radius of convergence is 0.
Example: find the radius of convergence of the sequence
The ratio test demands
for convergence so
As
for all
so the sequence converges for all
Example: find the radius of convergence of the sequence
The ratio test demands
for convergence so
The radius of convergence is 1.
