It's more of a confusion for me know: in what cases sums of infinite series can't be integrated and differentiated? I understand that e.g. $ \sum_{k=0}^{\infty}z^k = \frac{1}{1-z} $
has the remainder term of for $n+1, n+2...$ equal to $\frac{z^{n+1}}{1-z}$, so it does not converge uniformly for $|z|<1$. But nevertheless I can manipulate for $z<1$: $ \sum_{k=0}^{\infty}k z^k = z\sum_{k=0}^{\infty} \frac{\partial}{\partial z} (z^k) $
and so on. What do I miss here?