Is it possible to interchange differentiation and summation for infinite but also uniformly convergent sums, like: $\dfrac{d}{dx} \sin{x} = \dfrac{d}{dx}\sum_{n=0}^\infty (-1)^{n} \, \dfrac{x^{2n + 1}}{(2n +1)!} = \sum_{n=0}^\infty (-1)^{n} \, \dfrac{d}{dx} \dfrac{x^{2n + 1}}{(2n +1)!}$
Do you know how to show that the interchange of differentiation and summation is justified for uniformly convergent infinite sums? Or do you know any reference where this is stated?
