Theorem 1   If in the open interval $I$ , all the terms of the series
have continuous derivatives, the series converges having sum $S(x)$ and the differentiated series $f_1'(x)\!+\!f_2'(x)\!+\!\cdots$ 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 $$\frac{d\,S(x)}{dx} = f_1'(x)\!+\!f_2'(x)\!+\cdots$$ The situation implies also that the series (1) converges uniformly on $I$ .