linguistic:paraphrasing, linguistic:voice_change
Definition

For a power series f 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.