Region of convergence

In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges. It is either a non-negative real number or ${\displaystyle \infty }$. When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges.

For a power series ƒ defined as:

where

The radius of convergence r is a nonnegative real number or ${\displaystyle \infty }$ such that the series converges if

and diverges if

Some may prefer an alternative definition, as existence is obvious:

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.

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