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
and
diverges if
Example: find the radius of convergence of the sequence
in this example. The ratio test demands
for convergence so
Asso
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 demandsfor convergence so
Asfor all
so 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.