grandes-ecoles 2016 QI.C.2

grandes-ecoles · France · centrale-maths2__mp Reduction Formulae Bound or Estimate a Parametric Integral

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ Show that $$\left. \left. \forall (n, u) \in \mathbb{N}^{*} \times \right] 0, 1 \right], \quad \left| 1 - (\cos(\sqrt{2u/n}))^{n} \right| \leqslant u$$

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by
$$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$
Show that
$$\left. \left. \forall (n, u) \in \mathbb{N}^{*} \times \right] 0, 1 \right], \quad \left| 1 - (\cos(\sqrt{2u/n}))^{n} \right| \leqslant u$$

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.A.5 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2