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  $f_{1}^{{\prime}}(x)\!+\!f_{2}^{{\prime}}(x)\!+\!\cdots$  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

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