Convergence Theorem for Power Series
If
series diverges for
converges for
, then it converges absolutely for all
, then it diverges for all
. If the
DEFINITION: RADIUS AND INTERVAL OF CONVERGENCE
Possible behaviour of
.
1. There is an
(radius of convergence) such that the series diverges for
but
converges absolutely for
. The series may or may not converge at the endpoints
and
. Interval of Convergence :
or
2. The series converges absolutely for every . (
)
3. The series converges at
and diverges elsewhere.
Example. Determine the radius and interval of convergence of the power series
Solution:
Using Ratio test, we get
For the series to converge, we should have
as suggested by Ratio test. So
So, the radius of convergence
.
The interval of convergence is the solution set of
w/c is
included because they make the series a divergent p-series with
.
. End points are not