Let $A$ and $B$ be finite-dimensional Hermitian matrices, and let $v$ $:=$ $max${$||A||,||B||$}.
Show that $||(e^{-iAt/m}e^{-iBt/m})^m -e^{-i(A+B)t}||\leq\epsilon$ provided $m = \Omega(v^2t^2/\epsilon)$.
Let $A$ and $B$ be finite-dimensional Hermitian matrices, and let $v$ $:=$ $max${$||A||,||B||$}.
Show that $||(e^{-iAt/m}e^{-iBt/m})^m -e^{-i(A+B)t}||\leq\epsilon$ provided $m = \Omega(v^2t^2/\epsilon)$.
HINT: Consider the product of power series $\exp(A/m)\exp(B/m) = I + \frac{A}{m} + \frac{B}{m} + O\left(\frac{1}{m^2}\right)$. Then use this to estimate the logarithm for sufficiently large $m$, then exponentiate and raise to the $m$-th power.
This is from the proof in section 2.4 of "Lie Groups, Lie Algebras, and Representations" by Brian C. Hall.
As long as the limit definition of exponential function can be extended to operators, $$\lim_{N \rightarrow \infty }(1+\frac{\alpha}{N})^N=e^{\alpha}$$
$$ \alpha \in \mathbb{R} $$ $$ \alpha \in \operatorname{ \{Matrix,Operators, etc. ?\} } $$
then we can use it similarly
$$ \lim_{N \rightarrow \infty} \bigg( 1+ \frac{-it (A+B)}{N}+o(\frac{1}{N^2})\bigg)^N =e^{-it(A+B)} $$
Trotter formula, widely used in Path Integral, Quantum Monte-Carlo methods,