Why do we ever use the Dyson series? In dealing with interacting Hamiltonians, it's common to do expansions of some sort to make the problem tractable. Two common methods are the Dyson and Magnus expansion. Of the two, the Magnus expansion is quite interesting in that is preserves unitary time evolution at every order. This is not true for the Dyson series however. But, since this is such a nice property, my question is when and why would you prefer using the Dyson series over the Magnus expansion?
 A: *

*The Dyson and Magnus series are expanding $U(t_2,t_1)$ and $\ln U(t_2,t_1)$ wrt. the coupling constant $\lambda$, respectively. Here
$$\begin{align} U(t_2,t_1)~&=~\left\{\begin{array}{rcl}
T\exp\left[-\frac{i}{\hbar}\int_{t_1}^{t_2}\! dt~\lambda V(t)\right]
&\text{for}& t_1 ~<~t_2 \cr\cr
AT\exp\left[-\frac{i}{\hbar}\int_{t_1}^{t_2}\! dt~\lambda V(t)\right]
&\text{for}& t_2 ~<~t_1 \end{array}\right. \end{align}\tag{1}
$$
is the evolution operator, which satisfies two TDSEs
$$\begin{align} i\hbar \frac{\partial }{\partial t_2}U(t_2,t_1)
~=~&\lambda V(t_2)U(t_2,t_1),  \cr
i\hbar \frac{\partial }{\partial t_1}U(t_2,t_1)
~=~&-U(t_2,t_1)V(t_1)\lambda,\end{align}\tag{2} $$
along with the boundary condition
$$ U(t,t)~=~{\bf 1}.\tag{3}$$


*From the perspective of an all-order formal power series in $\lambda$, cf. perturbation theory, the Dyson series is much simpler than the Magnus series. (For starters, Wikipedia only provides a recursive formula for the latter.)


*We also note that the Dyson series is time-ordered (i.e. respects the time-ordering), while the Magnus series isn't. (The Magnus series contains commutators, which generically contain terms of opposite time-ordering.)


*However, as OP already mentions, in terms of a truncated expansion (in particular in numerical work), the Magnus expansion is often useful because of manifest unitarity in QM (which transcribes to a symplectic integrator in classical mechanics).
