Derivative of Uniformly Convergent Series of Continuously Differentiable Functions

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Theorem

Let $\left\langle{f_n}\right\rangle$ be a sequence of real functions.

Let each of $\left\langle{f_n}\right\rangle$ be continuously differentiable on the interval $\left[{a \,.\,.\, b}\right]$.

Let the series:

$\displaystyle f \left({x}\right) := \sum_{n \mathop = 1}^\infty f_n \left({x}\right)$

be pointwise convergent for all $x \in \left[{a \,.\,.\, b}\right]$.

Let the series:

$\displaystyle \sum_{n \mathop = 1}^\infty \frac {\mathrm d}{\mathrm dx} f_n \left({x}\right)$

be uniformly convergent for all $x \in \left[{a \,.\,.\, b}\right]$.


Then:

$\displaystyle \frac {\mathrm d}{\mathrm dx} f \left({x}\right) := \sum_{n \mathop = 1}^\infty \frac {\mathrm d}{\mathrm dx} f_n \left({x}\right)$


Proof


Sources