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Thus [ I_n = -\frac\cos nn + \frac\sin nn^2. ] As ( n \to \infty ), ( I_n = -\frac\cos nn + o\left(\frac1n\right) ). The amplitude of ( I_n ) is ( \sim \frac1n ) up to a bounded oscillatory factor. Indeed ( |I_n| \sim \fracn ), not ( C/n ) with constant sign, but in the sense of equivalence modulo ( o(1/n) ), it's ( O(1/n) ) and not ( o(1/n) ).
The book is a compilation of solved oral examination problems from the and the Écoles Normales Supérieures (ENS) , which are among the most selective higher education entrance exams in the world. Overview of "Analyse 4" Oraux X Ens Analyse 4 24.djvu
Take ( f(t) = t ). Then ( f(0)=0 ), ( f \in C^1 ). Thus [ I_n = -\frac\cos nn + \frac\sin nn^2
These oral sessions typically involve a "plateau"—a selection of exercises chosen by an examiner—that the student must solve on a blackboard in real-time. The topics range from standard applications to "extensions" that require genuine research skills. Indeed ( |I_n| \sim \fracn ), not (
If you are looking for a of such oral exams (volume 4, often covering integration, Fourier series, functional analysis, or differential equations), here is a sample problem in the style of "Oraux X-ENS Analyse" along with a full solution.
