# Weierstrass’ criterion of uniform convergence

###### Theorem.

Let the real functions ${f}_{1}(x)$, ${f}_{2}(x)$, … be defined in the interval $[a,b]$. If they all the condition

$$|{f}_{n}(x)|\leqq {M}_{n}\mathit{\hspace{1em}}\forall x\in [a,b],$$ |

with ${\sum}_{n=1}^{\mathrm{\infty}}{M}_{n}$ a convergent series^{} of , then the function series

$${f}_{1}(x)+{f}_{2}(x)+\mathrm{\cdots}$$ |

converges uniformly (http://planetmath.org/SumFunctionOfSeries) on the interval $[a,b]$.

The theorem is valid also for the series with complex function terms, when one replaces the interval with a subset of $\u2102$.

Title | Weierstrass’ criterion of uniform convergence^{} |
---|---|

Canonical name | WeierstrassCriterionOfUniformConvergence |

Date of creation | 2013-03-22 14:38:21 |

Last modified on | 2013-03-22 14:38:21 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 26A15 |

Classification | msc 40A30 |

Synonym | Weierstrass’ M-test |