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. Treat the previous question in the general case of a segment $[a,b]$ with $a < b$. One may examine $f_n \circ \sigma$ where $\sigma : [0,1] \rightarrow [a,b]$ is defined by $\sigma(t) = (1-t)a + tb$ for all $t \in [0,1]$.
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.
Treat the previous question in the general case of a segment $[a,b]$ with $a < b$. One may examine $f_n \circ \sigma$ where $\sigma : [0,1] \rightarrow [a,b]$ is defined by $\sigma(t) = (1-t)a + tb$ for all $t \in [0,1]$.