grandes-ecoles 2016 QI.C.4

grandes-ecoles · France · centrale-maths2__mp Reduction Formulae Determine Asymptotic Behavior or Limits of Sequences Defined by Integrals

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by $$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$ We admit the relation $\int_{0}^{\infty} \frac{\mathrm{e}^{-u}}{\sqrt{u}} \mathrm{~d}u = \sqrt{\pi}$.
Conclude that $u_{n} \sim \sqrt{\frac{n\pi}{2}}$.

We study the sequence $\left(u_{n}\right)_{n \in \mathbb{N}^{*}}$ defined by
$$\forall n \in \mathbb{N}^{*}, \quad u_{n} = \int_{0}^{\infty} \frac{1 - (\cos t)^{n}}{t^{2}} \mathrm{~d}t$$
We admit the relation $\int_{0}^{\infty} \frac{\mathrm{e}^{-u}}{\sqrt{u}} \mathrm{~d}u = \sqrt{\pi}$.

Conclude that $u_{n} \sim \sqrt{\frac{n\pi}{2}}$.

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