Give an example of an integer $K \in \mathbb{N}^\star$ for which the event $$\left\{\begin{array}{l}
\text{for all } k \in \llbracket 0, K \rrbracket, \text{ the function series } \sum X_n f_n^{(k)} \text{ is uniformly convergent on } [0,1], \\
\text{the function } \sum_{n=0}^{+\infty} X_n f_n \text{ is of class } \mathcal{C}^K, \\
\text{for all } k \in \llbracket 0, K \rrbracket, \left(\sum_{n=0}^{+\infty} X_n f_n\right)^{(k)} = \sum_{n=0}^{+\infty} X_n f_n^{(k)}
\end{array}\right\}$$ occurs with the functions $f_n$ defined by $$\left\{\begin{array}{l}
f_0 = 0 \\
f_n(x) = \ln\left(1 + \sin\left(\frac{x}{n}\right)\right) \quad \forall n \in \mathbb{N}^\star, \forall x \in [0,1].
\end{array}\right.$$
Give an example of an integer $K \in \mathbb{N}^\star$ for which the event
$$\left\{\begin{array}{l}
\text{for all } k \in \llbracket 0, K \rrbracket, \text{ the function series } \sum X_n f_n^{(k)} \text{ is uniformly convergent on } [0,1], \\
\text{the function } \sum_{n=0}^{+\infty} X_n f_n \text{ is of class } \mathcal{C}^K, \\
\text{for all } k \in \llbracket 0, K \rrbracket, \left(\sum_{n=0}^{+\infty} X_n f_n\right)^{(k)} = \sum_{n=0}^{+\infty} X_n f_n^{(k)}
\end{array}\right\}$$
occurs with the functions $f_n$ defined by
$$\left\{\begin{array}{l}
f_0 = 0 \\
f_n(x) = \ln\left(1 + \sin\left(\frac{x}{n}\right)\right) \quad \forall n \in \mathbb{N}^\star, \forall x \in [0,1].
\end{array}\right.$$