The Magnus expansion and Dyson series are very similar to each other, in that they both give a way to approximate a time-evolution operator as a series expansion
$$U(t) = \mathcal{T}\left(\exp\left[-i\int_0^tH(t')/\hbar\, dt'\right]\right)\tag{1}$$
where $\mathcal{T}$ is the time ordering operator$^1$. The Dyson series directly expands the above exponential as a sum of integrals, while the Magnus expansion finds a series expansion for $\bar{H}$ such that $$\exp(-i \bar{H} t/\hbar) = U(t).\tag{2}$$ The Magnus expansion expression for $U(t)$ is exactly unitary and has slightly better convergence properties, while the Dyson series is simpler.
In principle I can define a new kind of series for $U(t)$ by Taylor expanding the Magnus exponential like
$$U(t) = \exp(-i \bar{H}t/\hbar) \approx I -\frac{i \bar{H}t}{\hbar} - \frac{(\bar{H}t)^2}{2\hbar^2} + ...\tag{3}$$
Does this series expansion have a name? Does it have any advantages over a Dyson series?
Edit: I should clarify why one would use this expansion in practice, since it sounds silly on the surface; Oftentimes the Dyson and Magnus integrals are very complicated, so one does not want to go out many orders. This often precludes using a Dyson series, since it's especially important to preserve unitarity in numerical simulation. However exactly exponentiating the Magnus Hamiltonian $\bar{H}$ is expensive, so one often winds up Taylor expanding instead, per this question.
$^1$ Think of $\mathcal{T}$ as acting on each term in the Taylor series expansion of the exponential. It rearranges products of $H(t)$ such that later times are moved to the left of earlier times, e.g. $$\mathcal{T}(H(t_3)H(t_1)H(t_2) = H(t_3)H(t_2)H(t_1)\tag{4}$$ if $t_3>t_2>t_1$.
