In this part, we denote by $A$ and $B$ two arbitrary matrices in $\mathcal{M}_n(\mathbf{K})$. For every nonzero natural integer $k$, we set $$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$ We introduce the function $$\begin{aligned} h : \mathbf{R} & \longrightarrow \mathcal{M}_n(\mathbf{K}) \\ t & \longmapsto h(t) = e^{tA} e^{tB} - e^{t(A+B)} \end{aligned}$$ $\mathbf{7}$ ▷ Show that $$X_k - Y_k = O\left(\frac{1}{k^2}\right) \text{ as } k \rightarrow +\infty.$$
In this part, we denote by $A$ and $B$ two arbitrary matrices in $\mathcal{M}_n(\mathbf{K})$. For every nonzero natural integer $k$, we set
$$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
We introduce the function
$$\begin{aligned} h : \mathbf{R} & \longrightarrow \mathcal{M}_n(\mathbf{K}) \\ t & \longmapsto h(t) = e^{tA} e^{tB} - e^{t(A+B)} \end{aligned}$$
$\mathbf{7}$ ▷ Show that
$$X_k - Y_k = O\left(\frac{1}{k^2}\right) \text{ as } k \rightarrow +\infty.$$