# Cauchy’s root test

If $\sum {a}_{n}$ is a series of positive real terms and

$$ |

for all $n>N$, then $\sum {a}_{n}$ is convergent^{}. If $\sqrt[n]{{a}_{n}}\ge 1$ for an infinite number of values of $n$, then $\sum {a}_{n}$ is divergent.

## Limit form

Given a series $\sum {a}_{n}$ of complex terms, set

$$\rho =\underset{n\to \mathrm{\infty}}{lim\; sup}\sqrt[n]{|{a}_{n}|}$$ |

The series $\sum {a}_{n}$ is absolutely convergent if $$ and is divergent if $\rho >1$. If $\rho =1$, then the test is inconclusive.

Title | Cauchy’s root test |
---|---|

Canonical name | CauchysRootTest |

Date of creation | 2013-03-22 12:58:03 |

Last modified on | 2013-03-22 12:58:03 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 9 |

Author | Mathprof (13753) |

Entry type | Theorem |

Classification | msc 40A05 |

Synonym | root test |

Related topic | LambertSeries |