# Stolz-Cesaro theorem

Let ${({a}_{n})}_{n\ge 1}$ and ${({b}_{n})}_{n\ge 1}$ be two sequences of real numbers. If ${b}_{n}$ is positive, strictly increasing^{} and unbounded^{} and the following limit exists:

$$\underset{n\to \mathrm{\infty}}{lim}\frac{{a}_{n+1}-{a}_{n}}{{b}_{n+1}-{b}_{n}}=l$$ |

Then the limit:

$$\underset{n\to \mathrm{\infty}}{lim}\frac{{a}_{n}}{{b}_{n}}$$ |

also exists and it is equal to $l$.

Remark. This theorem is also valid if ${a}_{n}$ and ${b}_{n}$ are monotone^{}, tending to $0$.

Title | Stolz-Cesaro theorem |
---|---|

Canonical name | StolzCesaroTheorem |

Date of creation | 2013-03-22 13:17:16 |

Last modified on | 2013-03-22 13:17:16 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 9 |

Author | CWoo (3771) |

Entry type | Theorem |

Classification | msc 40A05 |

Related topic | CesaroMean |

Related topic | ExampleUsingStolzCesaroTheorem |

Related topic | KroneckersLemma |