Radius of Convergence - Convergence On The Boundary

Convergence On The Boundary

If the power series is expanded around the point a and the radius of convergence is r, then the set of all points z such that |z − a| = r is a circle called the boundary of the disk of convergence. A power series may diverge at every point on the boundary, or diverge on some points and converge at other points, or converge at all the points on the boundary. Furthermore, even if the series converges everywhere on the boundary (even uniformly), it does not necessarily converge absolutely.

Example 1: The power series for the function ƒ(z) = 1/(1 − z), expanded around z = 0, which is simply

has radius of convergence 1, and diverges at every point on the boundary.

Example 2: The power series for g(z) = −ln(1 − z), expanded around z = 0, which is

has radius of convergence 1, and diverges for z = 1 but converges for all other points on the boundary. The function ƒ(z) of Example 1 is the derivative of g(z).

Example 3: The power series

has radius of convergence 1 and converges everywhere on the boundary absolutely. If h is the function represented by this series on the unit disk, then the derivative of h(z) is equal to g(z)/z with g of Example 2. It turns out that h(z) is the dilogarithm function.

Example 4: The power series

has radius of convergence 1 and converges uniformly on the entire boundary {|z| = 1}, but does not converge absolutely on the boundary.