PlanetMath: termwise differentiation

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termwise differentiation

(Theorem)

Theorem 1 If in the open interval

, all the terms of the series

$\displaystyle f_1(x)\!+\!f_2(x)\!+\cdots$

(1)

and the differentiated series $f_1'(x)\!+\!f_2'(x)\!+\!\cdots$ converges uniformly on the interval

, then the series (1) can be differentiated termwise, i.e. in every point of

$\displaystyle \frac{d\,S(x)}{dx} = f_1'(x)\!+\!f_2'(x)\!+\cdots$

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

"termwise differentiation" is owned by Mathprof. [ owner history (1) ]

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See Also: power series

Other names:

differentiating a series

Keywords:

uniform convergence

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Cross-references: converges uniformly, implies, differentiable, sum function, point, interval, sum, converges, derivatives, continuous, series, open interval
There are 3 references to this entry.

This is version 6 of termwise differentiation, born on 2004-09-25, modified 2006-09-24.
Object id is 6231, canonical name is TermwiseDifferentiation.
Accessed 3612 times total.

Classification:

AMS MSC:	40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions)
	26A15 (Real functions :: Functions of one variable :: Continuity and related questions )

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