Convergence of Power Series

Theorem 6.5.6   Given a power series , there are exactly three possibilities:
1. The series conveges only when .
2. The series conveges for all .
3. There is an (called the radius of convergence'') such that converges for and diverges for .

Example 6.5.7   For the power series , the radius of convergence is .

Definition 6.5.8 (Radius of Convergence)   As mentioned in the theorem, is called the radius of convergence.

If the series converges only at , we say , and if the series converges everywhere we say that .

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.

Theorem 6.5.9   If has radius of convergence , then is differentiable on , and
1. ,
and both the derivative and integral have the same radius of convergence as .

Example 6.5.10   Find a power series representation for . Notice that

which has radius of convergence , since the above series is valid when , i.e., . Next integrating, we find that

for some constant . To find the constant, compute . We conclude that

Example 6.5.11   We will see later that the function has power series

Hence

This despite the fact that the antiderivative of is not an elementary function (see Example ).

William Stein 2006-03-15