Suppose we are given that a sequence of functions $f_n(z)$ convergences pointwise to $f(z)$ on the interval $[0,1]$. Suppose further that all of these functions are given by power series centered at 0 with radius of convergence $R > 1$.
To fix notation, say $f_n(z) = a_{n, 0} + a_{n, 1}z + a_{n, 2} z^2 + \dots$
and $f(z) = a_0 + a_1 z + a_2 z^2 + \dots$.
We are given $f_n(z) \to f(z)$ as $n \to \infty$, for each fixed $z \in [0,1]$.
Is it true that $a_{n, k} \to a_k$ as $n \to \infty$, for each $k$? Why? We can't use complex analysis it seems, since we are only on $[0,1]$.
If not, is it true under the additional assumption that all the $a_{n, k}$ and $a_n$ are uniformly bounded by some $M$?
No, this is false. Define $f_n(z) = \sin(nz)/n, n= 1,2,\dots$ Then $f_n(z) \to 0$ pointwise on $\mathbb R$ (in fact, uniformly on $\mathbb R.$) But $f_n'(0) = 1$ for all $n,$ while $f'(0) = 0.$