Editing Root test (section)

==Application to power series==

This test can be used with a [[power series]]

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

where the coefficients ''c''<sub>''n''</sub>, and the center ''p'' are [[complex number]]s and the argument ''z'' is a complex variable.

The terms of this series would then be given by ''a''<sub>''n''</sub> = ''c''<sub>''n''</sub>(''z'' &minus; ''p'')<sup>''n''</sup>. One then applies the root test to the ''a''<sub>''n''</sub> as above. Note that sometimes a series like this is called a power series "around ''p''", because the [[radius of convergence]] is the radius ''R'' of the largest interval or disc centred at ''p'' such that the series will converge for all points ''z'' strictly in the interior (convergence on the boundary of the interval or disc generally has to be checked separately).

A [[corollary]] of the root test applied to a power series is the [[Cauchy–Hadamard theorem]]: the radius of convergence is exactly <math>1/\limsup_{n \rightarrow \infty}{\sqrt[n]{|c_n|}},</math> taking care that we really mean ∞ if the denominator is&nbsp;0.