grandes-ecoles 2022 Q17

grandes-ecoles · France · centrale-maths2__official Sequences and Series Proof of Inequalities Involving Series or Sequence Terms

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 $x > 0$. Let $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$. Show that $|F(x)| \leqslant \frac{1}{3}$ for all $x \in [1,2]$.

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 $x > 0$. Let $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$. Show that $|F(x)| \leqslant \frac{1}{3}$ for all $x \in [1,2]$.

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