I'm trying to figure out the tightest upper bound for
$$\left \|\sum_{k = 0}^{N} \frac{(At)^k}{k!} \right\|,$$
where $A$ is a matrix and $\|A\|t > 1$. I know that $e^{\|A\|t}$ is a known upper bound, but I'm not sure if there's a tighter bound that depends on $N$.
I've tried the following:
$$ \left \|\sum_{k = 0}^{N} \frac{(At)^k}{k!} \right\| \leq \left \|(At)^N \sum_{k=0}^{N} \frac{1}{k!} \right\| \leq e(\|A\|t)^N, $$
but I'm wondering if there's something better I'm missing.