Editing Radius of convergence (section)

==Definition==
For a power series ''f'' defined as:

:<math>f(z) =  \sum_{n=0}^\infty c_n (z-a)^n, </math>

where

*''a'' is a  [[complex number|complex]] constant, the center of the [[disk (mathematics)|disk]] of convergence,
*''c''<sub>''n''</sub> is the ''n''-th complex coefficient, and
*''z'' is a complex variable.

The radius of convergence ''r'' is a nonnegative real number or <math>\infty</math> such that the series converges if

:<math>|z-a| < r</math>

and diverges if

:<math>|z-a| > r.</math>

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

: <math>r=\sup \left\{ |z-a|\ \left|\ \sum_{n=0}^\infty c_n(z-a)^n\ \text{ converges } \right.\right\} </math>

On the boundary, that is, where |''z''&nbsp;−&nbsp;''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 number]]s ''z''.<ref>{{Cite book|url=https://books.google.com/books?id=nw9eFnCSDNoC&q=radius+of+convergence|title=Mathematical Analysis-II|date=16 November 2010|publisher=Krishna Prakashan Media|language=en}}</ref>