I'm looking at an infinite series of continuous functions $\mathbb R \to \mathbb R$, say $\sum_{k=1}^\infty f_k(x)$, which converges pointwise to a function $f(x)$. Let $S_n$ be the n-th partial sum of the series.
If we assume that all $f_k$ are differentiable, I want the equation
$\frac{d}{dx} (\sum_{k=1}^\infty f_k(x)) = \sum_{k=1}^\infty f_k'(x)$
to be true for all $x$. Now we know that this is true, if $S'_n$ converges uniformly to $f'$. But what happens if we only have a weaker condition for $S'_n$:
For every $\epsilon>0$ there is a number $n$ such that $|f'-S'_n|<\epsilon$ holds for all $x$.
I highly appreciate your help and thoughts.