sum function of series

Let the terms of a series be real functions $f_n$ defined in a certain subset $A_0$ of $ℝ$; 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 $ℂ$).  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