# Cauchy criterion for convergence

A series ${\sum}_{i=0}^{\mathrm{\infty}}{a}_{i}$ in a Banach space $(V,\parallel \cdot \parallel )$ is http://planetmath.org/node/2311convergent^{} iff for every $\epsilon >0$ there is a number $N\in \mathbb{N}$ such that

$$ |

holds for all $n>N$ and $p\ge 1$.

## Proof:

First define

$${s}_{n}:=\sum _{i=0}^{n}{a}_{i}.$$ |

Now, since $V$ is complete^{}, $({s}_{n})$ converges^{} if and only if it is a Cauchy sequence^{}, so if for every $\epsilon >0$ there is a number $N$, such that for all $n,m>N$ holds:

$$ |

We can assume $m>n$ and thus set $m=n+p$. The series is iff

$$ |

Title | Cauchy criterion for convergence |
---|---|

Canonical name | CauchyCriterionForConvergence |

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

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

Owner | mathwizard (128) |

Last modified by | mathwizard (128) |

Numerical id | 14 |

Author | mathwizard (128) |

Entry type | Theorem |

Classification | msc 40A05 |