I want to know if the following is possible:
A sequence $f_n : \mathbb R \to \mathbb R$ of $C^\infty$ functions such that $$f := \sum_{n=1}^\infty f_n$$ converges uniformly to a $C^1$ function $f : \mathbb R \to \mathbb R$. So far so good but now the crazy stuff begins: I want $$f'(x) \ne \sum_{n=1}^\infty f_n'(x)$$ for all $x \in \mathbb R$. Either by don't letting $\sum_{n=1}^\infty f_n'(x)$ converge or by being convergent to a different number than $f'(x)$.
EDIT: I don't think that my question is simply about interchanging limits. I know that in general this is not allowed. But here we are asking for an example where it doesn't work for all (and not only for some) $x \in \mathbb R$.
