I am studying complex analysis, I knew that in $\mathbb{C}$, every convergent power series can do termwise integration or differentiation in their radii of convergence. And I vaguely remember that for a sequence of function, if $f_n\to f$ in $L^1$ on $I$, then
$$\lim_{n\to\infty}\int_If_ndx=\int_I\lim_{n\to\infty}f_ndx$$
What's the relationship between these two conditions?