Root test

In mathematics, the root test (sometimes called the Cauchy root test or Cauchy's radical test) is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity

${\displaystyle \limsup _{n\rightarrow \infty }{\sqrt[{n}]{|a_{n}|}},}$

where ${\displaystyle a_{n}}$ are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one.

The root test was developed first by Augustin-Louis Cauchy who published it in his textbook Cours d'analyse (1821).