Editing Root test (section)

== Root test explanation ==
[[File:Decision diagram for the root test.svg|thumb|Decision diagram for the root test]]
The root test was developed first by [[Augustin-Louis Cauchy]] who published it in his textbook [[Cours d'analyse]] (1821).<ref>{{citation|title=The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass|first=Umberto|last=Bottazzini|publisher=Springer-Verlag|year=1986|isbn=978-0-387-96302-0|pages=[https://archive.org/details/highercalculushi0000bott/page/116 116–117]|url=https://archive.org/details/highercalculushi0000bott/page/116}}. Translated from the Italian by Warren Van Egmond.</ref> Thus, it is sometimes known as the '''Cauchy root test''' or '''Cauchy's radical test'''. For a series

:<math>\sum_{n=1}^\infty a_n</math>

the root test uses the number

:<math>C = \limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|},</math>

where "lim sup" denotes the [[limit superior]], possibly +∞. Note that if

:<math>\lim_{n\rightarrow\infty}\sqrt[n]{|a_n|},</math>

converges then it equals ''C'' and may be used in the root test instead.

The root test states that:
* if ''C'' < 1 then the series [[converges absolutely]],
* if ''C'' > 1 then the series [[divergent series|diverges]],
* if ''C'' = 1 and the limit approaches strictly from above then the series diverges,
* otherwise the test is inconclusive (the series may diverge, converge absolutely or [[converge conditionally]]).

There are some series for which ''C'' = 1 and the series converges, e.g. <math>\textstyle \sum 1/{n^2}</math>, and there are others for which ''C'' = 1 and the series diverges, e.g. <math>\textstyle\sum 1/n</math>.