For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix $$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$ For $t \in \mathbb{R}$ and $n \in \mathbb{N}^{*}$, we introduce the matrix $$F_{n}(t) = I_{3} + \sum_{k=1}^{n} \frac{t^{k} \mathcal{M}^{k}}{k!} \in \mathcal{M}_{n}(\mathbb{K})$$ Show that the sequence $\left(F_{n}(t)\right)_{n \in \mathbb{N}}$ converges to a limit denoted $F(t) \in \mathcal{M}_{3}(\mathbb{R})$. Express $F(t)$ in terms of $R(\theta)$, where $\theta$ depends on $m$ and $t$.
For $m \in \mathbb{R}$, we denote by $\mathcal{M}$ the matrix
$$\mathcal{M} = \left(\begin{array}{ccc} 0 & -m & 0 \\ m & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)$$
For $t \in \mathbb{R}$ and $n \in \mathbb{N}^{*}$, we introduce the matrix
$$F_{n}(t) = I_{3} + \sum_{k=1}^{n} \frac{t^{k} \mathcal{M}^{k}}{k!} \in \mathcal{M}_{n}(\mathbb{K})$$
Show that the sequence $\left(F_{n}(t)\right)_{n \in \mathbb{N}}$ converges to a limit denoted $F(t) \in \mathcal{M}_{3}(\mathbb{R})$. Express $F(t)$ in terms of $R(\theta)$, where $\theta$ depends on $m$ and $t$.