Change of Summation and Differentiation

I am looking explicitly for a proof that that infinite summation of sequence of functions, uniformly convergent and differentiation are interchangable. $$\frac{\mathrm{d}}{\mathrm{d}x}\sum_{n=1}^{\infty}f_n(x)=\sum_{n=1}^{\infty}\frac{\mathrm{d}}{\mathrm{d}x}f_n(x)$$ and an explicit example where pointwise convergent series fail to do so.

THIS post has same question but no proof. The refered link in 3rd edition of rudin has related but different proof.

Solution 1:

The classical theorem is that if each $f_n$ and $f_n'$ are continuous on an interval, the series $$ f(x) = \sum_{n=1}^\infty f_n(x) $$ converges pointwise, and the series $$ g(x) = \sum_{n=1}^\infty f_n'(x) $$ converges uniformly, then $f$ is differentiable and $f' = g$. This uses the fundamental theorem of calculus and the ability to interchange uniform limits with integrals.