Root test

In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series

:$sum\_\{n=1\}^infty\; a\_n.$

It is particularly useful in connection with power series.

The test

The root test was developed first by Cauchy and so is sometimes known as the Cauchy root test or Cauchy's radical test. The root test uses the number

:$C\; =\; limsup\_\{n\; ightarrowinfty\}sqrt\; [n]\; .$

If

equivalent to

:$sqrt\; [n]\; =\; sqrt\; [n]\; =\; 1,$

(since points > 1 will diverge) and this will not change the radius of convergence since these are just the points lying on the boundary of the interval or disc, so

:$R\; =\; 1/limsup\_\{n\; ightarrow\; infty\}\{sqrt\; [n]\; \{|c\_n|.$

ee also

* Ratio test
* Convergent series

References

* cite book
author= Knopp, Konrad
title= Infinite Sequences and Series
chapter = § 3.2
publisher=Dover publications, Inc., New York
year=1956
id = ISBN 0-486-60153-6
* cite book
author= Whittaker, E. T., and Watson, G. N.
title= A Course in Modern Analysis
chapter = § 2.35
edition=fourth edition
publisher=Cambridge University Press
year=1963
id = ISBN 0-521-58807-3