termwise differentiation

Theorem.

If in the open interval $I$, all the of the series

$f_1(x)+f_2(x)+\mathrm{\cdots }$ (1)

have continuous derivatives, the series converges having sum $S(x)$ and the differentiated series  $f_1^{\prime }(x)+f_2^{\prime }(x)+\mathrm{\cdots }$  converges uniformly (http://planetmath.org/SumFunctionOfSeries) on the interval $I$, then the series (1) can be differentiated termwise, i.e. in every point of $I$ the sum function $S(x)$ is differentiable and

$$\frac{dS(x)}{dx}=f_1^{\prime }(x)+f_2^{\prime }(x)+\mathrm{\cdots }$$

The situation implies also that the series (1) converges uniformly on $I$.

Title	termwise differentiation
Canonical name	TermwiseDifferentiation
Date of creation	2013-03-22 14:38:38
Last modified on	2013-03-22 14:38:38
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	9
Author	Mathprof (13753)
Entry type	Theorem
Classification	msc 26A15
Classification	msc 40A30
Synonym	differentiating a series
Related topic	PowerSeries
Related topic	IntegrationOfLaplaceTransformWithRespectToParameter
Related topic	IntegralOfLimitFunction