Radius of Convergence - Definition

Definition

For a power series ƒ defined as:

where

a is a complex constant, the center of the disk of convergence,

c_n is the nth complex coefficient, and

z is a complex variable.

The radius of convergence r is a nonnegative real number or ∞ such that the series converges if

and diverges if

In other words, the series converges if z is close enough to the center and diverges if it is too far away. The radius of convergence specifies how close is close enough. On the boundary, that is, where |z − a| = r, the behavior of the power series may be complicated, and the series may converge for some values of z and diverge for others. The radius of convergence is infinite if the series converges for all complex numbers z.