# 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

 $\displaystyle f_{1}+f_{2}+\cdots$ (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$S_{1},\,S_{2},\,\ldots$  of the partial sums$S_{n}=f_{1}\!+\!f_{2}\!+\cdots+\!f_{n}$  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 $\varepsilon$ there is an integer $n_{\varepsilon}$ such that each value of $x$ in the interval  $[a,\,b]$  the inequality

 $|S_{n}(x)-S(x)|<\varepsilon$

when  $n\geqq n_{\varepsilon}$.

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 $\mathbb{C}$).  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