Let $T(t),t\geq 0$, be a $C_0$-semigroup on a Banach space $X$. If $A$ is the infinitesimal generator of $T(t),t\geq 0$, then $$T(t)x=\lim_{n\infty}(I-\frac{t}{n}A)^{-n}x$$ for every $x \in X, t\geq 0$, and the limit is uniform on any bounded interval $[a,b]\subset [0,\infty)$. This is the exponential formula.
I wonder if one could affirm that the sequence $(I-\frac{t}{n}A)^{-n}$ converges in norm to $T(t)$ and the convergence is uniform on any bounded interval $[a,b]\subset [0,\infty)$?