grandes-ecoles 2022 Q15

grandes-ecoles · France · centrale-maths2__official Proof Proof That a Map Has a Specific Property

For all $n \in \mathbb{N}^\star$, let $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfy $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$. Show that the function series $\sum f_n(x)$ converges normally on any segment contained in $]0,+\infty[$ and that the function $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$ is of class $\mathcal{C}^2$ on $]0,+\infty[$.

For all $n \in \mathbb{N}^\star$, let $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfy $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$. Show that the function series $\sum f_n(x)$ converges normally on any segment contained in $]0,+\infty[$ and that the function $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$ is of class $\mathcal{C}^2$ on $]0,+\infty[$.

Paper Questions

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