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 $$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$ The objective is to prove the relation $$\lim_{k \rightarrow +\infty} \left(e^{\frac{A}{k}} e^{\frac{B}{k}}\right)^k = e^{A+B}.$$ $\mathbf{8}$ ▷ Verify the relation $$X_k^k - Y_k^k = \sum_{i=0}^{k-1} X_k^i \left(X_k - Y_k\right) Y_k^{k-i-1}$$ Deduce from this the relation $\lim_{k \rightarrow +\infty} \left(e^{\frac{A}{k}} e^{\frac{B}{k}}\right)^k = e^{A+B}$.
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
$$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
The objective is to prove the relation
$$\lim_{k \rightarrow +\infty} \left(e^{\frac{A}{k}} e^{\frac{B}{k}}\right)^k = e^{A+B}.$$
$\mathbf{8}$ ▷ Verify the relation
$$X_k^k - Y_k^k = \sum_{i=0}^{k-1} X_k^i \left(X_k - Y_k\right) Y_k^{k-i-1}$$
Deduce from this the relation $\lim_{k \rightarrow +\infty} \left(e^{\frac{A}{k}} e^{\frac{B}{k}}\right)^k = e^{A+B}$.