sum function of series
Let the terms of a series be real functions defined in a certain subset of ; we can speak of a function series. All points where the series
converges form a subset of , and we have the of (1) defined in .
If the sequence of the partial sums of the series (1) converges uniformly (http://planetmath.org/LimitFunctionOfSequence) in the interval to a function , 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 having sum function , in the interval , if for every positive number there is an integer such that each value of in the interval the inequality
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 continuous term-functions is continuous and may be integrated termwise.
|Title||sum function of series|
|Date of creation||2013-03-22 14:38:15|
|Last modified on||2013-03-22 14:38:15|
|Last modified by||pahio (2872)|
|Defines||uniform convergence of series|