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Hometermwise differentiation

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# termwise differentiation

###### Theorem.

If in the open interval $I$, all the terms 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 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 $I$.

Keywords:

uniform convergence

Related:

PowerSeries, IntegrationOfLaplaceTransformWithRespectToParameter, IntegralOfLimitFunction

Synonym:

differentiating a series

Major Section:

Reference

Type of Math Object:

Theorem

Parent:

## Mathematics Subject Classification

26A15*no label found*40A30

*no label found*

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