Let $K \in \mathbb{N}^\star$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying: (H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$; (H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent. According to the result of the previous question, we can set $F_k(x) = \sum_{n=0}^{+\infty} f_n^{(k)}(x)$ for all $x \in [a,b]$. Prove that $F_0$ is of class $\mathcal{C}^K$ on $[a,b]$ and that $F_0^{(k)} = F_k$ for all $k \in \llbracket 1, K \rrbracket$.
Let $K \in \mathbb{N}^\star$, consider distinct real numbers $x_1 < \cdots < x_K$ in an interval $[a,b]$ (with $a < b$), and a sequence of functions $(f_n)$ of class $\mathcal{C}^K$ on $[a,b]$ with real values satisfying:
(H1) the function series $\sum f_n^{(K)}$ converges normally on $[a,b]$;
(H2) for all $\ell \in \llbracket 1, K \rrbracket$ the numerical series $\sum f_n(x_\ell)$ is absolutely convergent.
According to the result of the previous question, we can set $F_k(x) = \sum_{n=0}^{+\infty} f_n^{(k)}(x)$ for all $x \in [a,b]$. Prove that $F_0$ is of class $\mathcal{C}^K$ on $[a,b]$ and that $F_0^{(k)} = F_k$ for all $k \in \llbracket 1, K \rrbracket$.