# sum function of series

Let the terms of a series be real functions ${f}_{n}$ defined in a certain subset ${A}_{0}$ of $\mathbb{R}$; we can speak of a function series. All points $x$ where the series

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

converges^{} form a subset $A$ of ${A}_{0}$, and we have the $S:x\mapsto S(x)$ of (1) defined in $A$.

If the sequence ${S}_{1},{S}_{2},\mathrm{\dots}$ of the partial sums ${S}_{n}={f}_{1}+{f}_{2}+\mathrm{\cdots}+{f}_{n}$ of the series (1) converges uniformly (http://planetmath.org/LimitFunctionOfSequence) in the interval $[a,b]\subseteq A$ to a function $S:x\mapsto S(x)$, we say that the series in this interval. We may also set the direct

Definition. The function series (1), which converges in every point of the interval $[a,b]$ having sum function $S:x\mapsto S(x)$,
in the interval $[a,b]$, if for every positive number $\epsilon $ there is an integer ${n}_{\epsilon}$ such that each value of $x$ in the interval $[a,b]$ the inequality^{}

$$ |

when $n\geqq {n}_{\epsilon}$.

Note. One can without trouble be convinced that the term functions of a uniformly converging series converge uniformly to 0 (cf. the necessary condition of convergence).

The notion of of series can be extended to the series with complex function terms (the interval is replaced with some subset of $\u2102$). The significance of the is therein that the sum function of a series with this property and with continuous^{} term-functions is continuous and may be integrated termwise.

Title | sum function of series |

Canonical name | SumFunctionOfSeries |

Date of creation | 2013-03-22 14:38:15 |

Last modified on | 2013-03-22 14:38:15 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 18 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 26A15 |

Classification | msc 40A30 |

Related topic | UniformConvergenceOfIntegral |

Related topic | SumOfSeries |

Related topic | OneSidedContinuityBySeries |

Defines | function series |

Defines | sum function |

Defines | uniform convergence^{} of series |