Radius of Convergence - A Graphical Example

A Graphical Example

Consider the function 1/(z2 + 1).

This function has poles at z = ±i.

As seen in the first example, the radius of convergence of this function's series in powers of (z − 0) is 1, as the distance from 0 to each of those poles is 1.

Then the Taylor series of this function around z = 0 will only converge if |z| < 1, as depicted on the example on the right.