# Weierstrass M-test

Let $X$ be any set, ${\{{f}_{n}\}}_{n\in \mathbb{N}}$ a sequence of real or complex valued functions on $X$ and ${\{{M}_{n}\}}_{n\in \mathbb{N}}$ a sequence of non-negative real numbers. Suppose that, for each $n\in \mathbb{N}$ and $x\in X$, we have $|{f}_{n}(x)|\le {M}_{n}$. Then $f={\sum}_{n=1}^{\mathrm{\infty}}{f}_{n}$ converges uniformly if ${\sum}_{n=1}^{\mathrm{\infty}}{M}_{n}$ converges.

Title | Weierstrass M-test^{} |
---|---|

Canonical name | WeierstrassMtest |

Date of creation | 2013-03-22 12:56:11 |

Last modified on | 2013-03-22 12:56:11 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 13 |

Author | yark (2760) |

Entry type | Theorem |

Classification | msc 30A99 |

Related topic | AbsoluteConvergence |