**Question:** Given $f\in C^{\infty}$ which is *not* analytic on a bounded domain $\Omega \subseteq \mathbb{R}$. What can we say about the sequence $\lbrace f^{(m)} \rbrace _{m=1}^{\infty} $? Specifically - what can we say about the convergence and/or decay of its $L^2$ and $L^{\infty}$ norms? If nothing, why? Of special importance is to bound globally Taylor like terms of the form $f^{(m)}(\xi)/m! $ for some $\xi \in \Omega$.

*Motivation, not Neccessary:* My question is motivated by the residue terms of numerical integration formulas. We can usually have nice convergence rates for analytical functions due to the exponential decay in their taylor series coefficients $f^{(m)}/m! $.

I have read some of the math.stackexchange posts about it, and while this one does contain some possible leads, I couldn't find the answers to my questions there yet.

Thanks