Given a power series
there are exactly three possibilities:
- The series conveges only when .
- The series conveges for all .
- There is an (called the ``radius of convergence'')
and diverges for .
For the power series
, the radius
of convergence is
If the series converges only at , we say , and if the series
converges everywhere we say that .
(Radius of Convergence)
As mentioned in the theorem,
is called the radius of convergence
The interval of convergence is the set of for which the
series converges. It will be one of the following:
The point being that the statement of the theorem only asserts
something about convergence of the series on the open interval
. What happens at the endpoints of the interval is
not specified by the theorem; you can only figure it out by
looking explicitly at a given series.
has radius of convergence ,
and both the derivative and integral have the same radius of convergence as .
Find a power series representation for
which has radius of convergence
, since the above series
is valid when
Next integrating, we find that
for some constant
To find the constant, compute
We conclude that
We will see later that the function
has power series
This despite the fact that the antiderivative of
is not an
elementary function (see Example