sum function of series

Let the terms of a series be real functions fn defined in a certain subset A0 of ; we can speak of a function series.  All points x where the series

f1+f2+ (1)

convergesPlanetmathPlanetmath form a subset A of A0, and we have the   S:xS(x)  of (1) defined in A.

If the sequenceS1,S2,  of the partial sumsSn=f1+f2++fn  of the series (1) converges uniformly ( in the interval[a,b]A  to a functionS:xS(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:xS(x),  in the interval  [a,b],  if for every positive number ε there is an integer nε such that each value of x in the interval  [a,b]  the inequalityMathworldPlanetmath


when  nnε.

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 ).  The significance of the is therein that the sum function of a series with this property and with continuousMathworldPlanetmathPlanetmath 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 convergenceMathworldPlanetmath of series