Oraux X Ens Analyse 4 24.djvu -

I cannot directly access external files such as Oraux X Ens Analyse 4 24.djvu . However, if you provide the text or a specific exercise from that document (e.g., by copying the statement or describing the problem), I can certainly help produce a detailed solution, commentary, or a synthetic correction typical of an oral examination at ENS/X level in analysis.

Better: By Riemann–Lebesgue lemma, for any ( g \in L^1 ), ( \int g(t) \cos(nt) dt \to 0 ). Here ( g = f' \in L^1 ). Therefore [ \int_0^1 f'(t) \cos(nt) , dt \to 0. ] Hence [ I_n = \frac1n \cdot o(1) = o\left(\frac1n\right). ] Example with ( I_n \sim C/n ) Take ( f(t) = t ). Then ( f(0)=0 ), ( f \in C^1 ). Oraux X Ens Analyse 4 24.djvu

The integral term: ( \left| \int_0^1 f'(t) \cos(nt) , dt \right| \leq \int_0^1 |f'(t)| dt < \infty ), hence it is bounded. Thus the whole integral term is ( O(1/n) ). Wait — but we need ( o(1/n) ), not just ( O(1/n) ). I cannot directly access external files such as