Series of functions and differentiating term by term

[mahler1]

I have to solve a bunch of exercises related to function series and in some of them they ask me whether a particular series converges uniformly if one differentiates it term by term. So here I came up with a doubt: When $\sum_0^{\infty} f_n(x)$ can be differentiated term by term? What hypothesis do I need to do that? Does the series have to converge to a differentiable function or I can differentiate each term without even asking the series to be convergent? I am very confused with this.

 

[iRaid]

https://www.dpmms.cam.ac.uk/~agk22/uniform.pdf
Might be what you're looking for?

 

[mahler1]

https://www.dpmms.cam.ac.uk/~agk22/uniform.pdf
Might be what you're looking for?

Yes, thanks.