page_title: Root test
section_title: Root test explanation
hierarchical_section_title: Root test / Root test explanation
context_page_description: In mathematics, the root test is a criterion for the convergence of an infinite series. It depends on the quantity
  where 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. It is particularly useful in connection with power series.
context_section_description: The root test was developed first by Augustin-Louis Cauchy who published it in his textbook Cours d'analyse (1821). Thus, it is sometimes known as the Cauchy root test or Cauchy's radical test. For a series
  the root test uses the number
  where "lim sup" denotes the limit superior, possibly ∞+.  Note that if
  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 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. , and there are others for which C = 1 and the series diverges, e.g. .
caption: Decision diagram for the root test