# termwise differentiation

###### Theorem.

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

 $\displaystyle f_{1}(x)\!+\!f_{2}(x)\!+\cdots$ (1)

have continuous derivatives, the series converges having sum $S(x)$ and the differentiated series  $f_{1}^{\prime}(x)\!+\!f_{2}^{\prime}(x)\!+\!\cdots$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

 $\frac{d\,S(x)}{dx}=f_{1}^{\prime}(x)\!+\!f_{2}^{\prime}(x)\!+\cdots$

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

Title termwise differentiation TermwiseDifferentiation 2013-03-22 14:38:38 2013-03-22 14:38:38 Mathprof (13753) Mathprof (13753) 9 Mathprof (13753) Theorem msc 26A15 msc 40A30 differentiating a series PowerSeries IntegrationOfLaplaceTransformWithRespectToParameter IntegralOfLimitFunction