Editing Radius of convergence (section)

===Theoretical radius===

The radius of convergence can be found by applying the [[root test]] to the terms of the series. The root test uses the number

:<math>C = \limsup_{n\to\infty}\sqrt[n]{|c_n(z-a)^n|} = \limsup_{n\to\infty} \left(\sqrt[n]{|c_n|}\right) |z-a|</math>

"lim&nbsp;sup" denotes the [[limit superior]].  The root test states that the series converges if ''C''&nbsp;<&nbsp;1 and diverges if&nbsp;''C''&nbsp;>&nbsp;1.  It follows that the power series converges if the distance from ''z'' to the center ''a'' is less than

:<math>r = \frac{1}{\limsup_{n\to\infty}\sqrt[n]{|c_n|}}</math>

and diverges if the distance exceeds that number; this statement is the [[Cauchy–Hadamard theorem]].  Note that ''r''&nbsp;=&nbsp;1/0 is interpreted as an infinite radius, meaning that ''f'' is an [[entire function]].

The limit involved in the [[ratio test]] is usually easier to compute, and when that limit exists, it shows that the radius of convergence is finite.

<!-- NOTE: The ratio test as usually stated involves c_{n+1}/c_n, but THIS statement correctly uses c_{n+1}/c_n.  -->
:<math>r = \lim_{n\to\infty} \left| \frac{c_{n}}{c_{n+1}} \right|.</math>
<!-- NOTE: The ratio test as usually stated involves c_{n+1}/c_n, but THIS statement correctly uses c_n/c_{n+1}. -->

This is shown as follows.  The ratio test says the series converges if

: <math> \lim_{n\to\infty} \frac{|c_{n+1}(z-a)^{n+1}|}{|c_n(z-a)^n|} < 1. </math>

That is equivalent to

: <math> |z - a| < \frac{1}{\lim_{n\to\infty} \frac{|c_{n+1}|}{|c_n|}} = \lim_{n\to\infty} \left|\frac{c_n}{c_{n+1}}\right|. </math>